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On the Optimal Boundary of a Three-Dimensional

Singular Stochastic Control Problem Arising in

Irreversible Investment∗

Tiziano De Angelis† Salvatore Federico‡ Giorgio Ferrari§

June 20, 2014

Abstract. This paper examines a Markovian model for the optimal irreversible investment

problem of a firm aiming at minimizing total expected costs of production. We model market

uncertainty and the cost of investment per unit of production capacity as two independent

one-dimensional regular diffusions, and we consider a general convex running cost function.

The optimization problem is set as a three-dimensional degenerate singular stochastic control

problem.

We provide the optimal control as the solution of a Skorohod reflection problem at a suitable

free-boundary surface. Such boundary arises from the analysis of a family of two-dimensional

parameter-dependent optimal stopping problems and it is characterized in terms of the family of

unique continuous solutions to parameter-dependent nonlinear integral equations of Fredholm

type.

Key words: irreversible investment, singular stochastic control, optimal stopping, free-

boundary problems, nonlinear integral equations.

MSC2010: 93E20, 60G40, 35R35, 91B70.

JEL classification: C02, C73, E22, D92.

∗The first author was supported by EPSRC grant EP/K00557X/1; Financial support by the German Research

Foundation (DFG) via grant Ri–1128–4–1 is gratefully acknowledged by the third author. This work was started

during a visit of the second author at the Center for Mathematical Economics (IMW) at Bielefeld University thanks

to a grant by the German Academic Exchange Service (DAAD). The second author thankfully aknowledges the

financial support by DAAD and the hospitality of IMW.†School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom;

tiziano.deangelis@manchester.ac.uk‡Dipartimento di Economia, Management e Metodi Quantitativi, Universita degli Studi di Milano, Via Con-

servatorio 7, 20122 Milano, Italy; salvatore.federico@unimi.it§Center for Mathematical Economics (IMW), Bielefeld University, Universitatsstrasse 25, D-33615 Bielefeld,

Germany; giorgio.ferrari@uni-bielefeld.de

1

The Optimal Boundary of an Irreversible Investment Problem 2

1 Introduction

In this paper we study a Markovian model for a firm’s optimal irreversible investment problem.

The firm aims at minimizing total expected costs of production when its running cost function

depends on the uncertain condition of the economy as well as on on the installed production

capacity, and the cost of investment per unit of production capacity is random. In mathematical

terms, this amounts to solving the three-dimensional degenerate singular stochastic control

problem

V (x, y, z) := infνE

[ ∫ ∞

0e−rtc(Xx

t , z + νt)dt+

∫ ∞

0e−rtY y

t dνt

], (1.1)

where the infimum is taken over a suitable set of nondecreasing admissible controls. Here X

and Y are two independent one-dimensional diffusion processes modeling market uncertainty

and the cost of investment per unit of production capacity, respectively. The control process νtis the cumulative investment made up to time t and c is a general convex cost function. We

solve problem (1.1) by relying on the connection existing between singular stochastic control and

optimal stopping (see, e.g., [1] and [26]). In fact, we provide the optimal investment strategy

ν∗ in terms of a free-boundary surface (x, y) 7→ z∗(x, y) that splits the state space into action

and inaction regions. Such surface arises from an associated family of two-dimensional, infinite

time-horizon optimal stopping problems and it is uniquely characterized through a family of

continuous solutions to parameter-dependent, nonlinear integral equations of Fredholm type.

To the best of our knowledge this is a new feature in the theory of singular stochastic control of

multi-dimensional systems.

The connection between singular stochastic control and optimal stopping has been thoro-

ughly studied in the literature. It turns out that under appropriate assumptions the derivative

of V in the direction of the controlled state variable equals the value function of a suitable

optimal stopping problem whose first optimal stopping time is τ∗ = inft ≥ 0 : ν∗t > 0, with

ν∗ the optimal control (see, e.g., [26]). This feature was firstly noticed in [4] and then it was

rigorously proved, via purely probabilistic arguments, in [26] in the case of a Brownian motion

additively controlled by a nondecreasing process. Later on, this kind of link was established

also for more complicated dynamics of the controlled diffusion (see, e.g., [1], [5], and [6]) and,

recently, singular stochastic control problems with controls of bounded-variation were brought

in contact with zero-sum optimal stopping games in [7] and [28].

In the mathematical economic literature singular stochastic control problems are often em-

ployed to model the irreversible (partially reversible) optimal investment problem of a firm op-

erating in an uncertain environment (see [11], [13], [18], [19], [24], [29], [33], [39] and references

therein, among many others). The monotone (bounded-variation) control represents in fact the

cumulative investment (investment-disinvestment) policy of such firm its aim is maximizing to-

tal net expected profits or, alternatively, minimizing total expected costs. The optimal timing

problem associated to the optimal investment one is then related to real options as pointed out

by [32] and [37] among others.

The Optimal Boundary of an Irreversible Investment Problem 3

Problems of stochastic irreversible (or partially reversible) investment have been tackled via a

number of different approaches. Among others, these include dynamic programming techniques

(see, e.g., [18], [24], [29] and [33]), stochastic first-order conditions and the Bank-El Karoui’s

Representation Theorem [2] (see, e.g., [3], [12], [19] and [39]).

Notice that due to the three-dimensional structure of our problem (1.1) a direct study of the

associated Hamilton-Jacobi-Bellman equation with the aim of finding explicit smooth solutions

(as in the two-dimensional problem of [33], among others) seems hard to apply. In fact, differently

to, e.g., [33], in our case the linear part of the Hamilton-Jacobi-Bellman equation for the value

function of problem (1.1) is a PDE (rather than a ODE) and it does not have a general solution.

On the other hand, arguing as in [19], we might tackle problem (1.1) by relying on a stochastic

first-order conditions approach; that would allow us to characterize the unique optional solution

l∗ of the Bank-El Karoui representation problem (cf. [2]) as l∗t = z∗(Xxt , Y

yt ), with z

∗ the free-

boundary surface that splits the state space into action and inaction regions. However, the

integral equation for the free-boundary which derives from the main result of [19] (i.e., [19,

Th. 3.11]) cannot be found in our multi-dimensional setting. Therefore it seems very hard to

obtain any information on the geometry of the free-boundary surface z∗(x, y) by using only the

characterization of the process l∗t .

In this paper we study problem (1.1) by relying on the connection between singular stochastic

control and optimal stopping and by combining techniques from probability and PDE theory. We

show that the optimal control ν∗ is the minimal effort needed to keep the (optimally controlled)

state process above a free-boundary surface z∗ whose level curves z∗(x, y) = z, z ∈ R+, are

the free-boundaries y∗( · ; z) of the parameter-dependent optimal stopping problems associated

to the original singular control one. Under some further mild conditions, we characterize each

optimal boundary y∗( · ; z), z ∈ R+, as the unique continuous solution of nonlinear integral

equation of Fredholm type (see our Theorem 4.10 below).

The issue of finding integral equations for the free-boundary of optimal stopping problems

has been successfully addressed in a number of papers (cf. [35] for a survey). In the context of

one-dimensional stochastic (ir)reversible investment problems on a finite time-horizon integral

equations for the optimal boundaries have been obtained by an application of Peskir’s local time-

space calculus (see [11] and [13] and references therein for details). However, those arguments

cannot be applied in our case since it seems quite hard to prove that the process y∗(Xxt ; z), t ≥

0 is a semimartingale for each given z ∈ R+ as it is required in [36, Th. 2.1]. On the other

hand, multi-dimensional settings have been studied for instance in [35, Sec. 13] where a diffusion

X was considered along with its running supremum S. Unlike [35, Sec. 13] here we deal with

a genuine two dimensional diffusion (X,Y ) with X and Y independent. This gives rise to a

completely different analysis of the problem and new methods have been developed.

The paper is organized as follows. In Section 2 we set the stochastic irreversible investment

problem. In Sections 3 and 4 we introduce the associated family of optimal stopping problems

and we characterize its value functions and its optimal-boundaries. The form of the optimal

control is provided in Section 5. Finally, some technical results are discussed in Appendix A.

The Optimal Boundary of an Irreversible Investment Problem 4

2 The Stochastic Irreversible Investment Problem

In this section we set the stochastic irreversible investment problem object of our study. Let

(Ω,F , (Ft)t≥0,P) be a complete filtered probability space with F = Ft, t ≥ 0 the filtration

generated by a two-dimensional Brownian motion W = (W 1t ,W

2t ), t ≥ 0 and augmented with

P-null sets.

1. A real process X = Xt, t ≥ 0 represents the uncertain status of the economy (typically,

the demand of a good or, more generally, some indicator of macroeconomic conditions). We

assume thatX is a time-homogeneous Markov diffusion satisfying the stochastic differential

equation (SDE)

dXt = µ1(Xt)dt+ σ1(Xt)dW1t , X0 = x, (2.1)

for some Borel functions µ1 and σ1 to be specified. To account for the dependence of X

on its initial position we denote the solution of (2.1) by Xx.

2. A one-dimensional positive process Y = Yt, t ≥ 0 represents the cost of investment per

unit of production capacity. We assume that Y evolves according to the SDE

dYt = µ2(Yt)dt+ σ2(Yt)dW2t , Y0 = y, (2.2)

for some Borel functions µ2 and σ2 to be specified as well. Again, to account for the

dependence of Y on y, we denote the solution of (2.2) by Y y.

3. A control process ν = νt, t ≥ 0 describes an investment policy of the firm and νt is the

cumulative investment made up to time t. We say that a control process ν is admissible if

it belongs to the nonempty convex set

V := ν : Ω×R+ 7→ R

+ | t 7→ νt is cadlag, nondecreasing, F-adapted. (2.3)

In the following we set ν0− = 0, for every ν ∈ V.

4. A purely controlled process Z = Zt, t ≥ 0, represents the production capacity of the

firm and it is defined by

Zt := z + νt, z ∈ R+. (2.4)

The process Z depends on its initial position z and on the control (investment) process ν,

therefore we denote it by Zz,ν.

We assume that the uncontrolled diffusions Xx and Y y have state-space I1 = (x, x) ⊆ R

and I2 = (y, y) ⊆ R+, respectively, with x, x, y, y natural boundary points. We recall that a

boundary point ξ is natural for one of our diffusion processes if it is: non-entrance and non-exit.

That is, ξ cannot be a starting point for the process and it cannot be reached in finite time

The Optimal Boundary of an Irreversible Investment Problem 5

(cf. for instance [9, Ch. 2, p. 15]). Moreover if such ξ is finite one also has µi(ξ) = σi(ξ) = 0

with i = 1 if ξ = x (or ξ = x) and with i = 2 if ξ = y (or ξ = y). That is shown in Appendix

A.1 for the sake of completeness.

We make the following

Assumption 2.1.

(i) The coefficients µi : R 7→ R, σi : R 7→ R+, i = 1, 2, are such that

|µi(ζ)− µi(ζ′)| ≤ Ki|ζ − ζ ′|, |σi(ζ)− σi(ζ

′)| ≤ Mi|ζ − ζ ′|γ , ∀ζ, ζ ′ ∈ Ii,

for some Ki > 0, Mi > 0 and γ ∈ [12 , 1].

(ii) The diffusions Xx and Y y are nondegenerate, i.e. σ2i > 0 in Ii, i = 1, 2.

Assumption 2.1 guarantees that

∫ ζ+εo

ζ−εo

1 + |µi(y)|

|σi(y)|2dy < +∞, for some εo > 0 and every ζ in Ii (2.5)

and hence both (2.1) and (2.2) have a weak solution that is unique in the sense of probability

law (cf. [27, Ch. 5.5]). Such solutions do not explode in finite time due to the sublinear growth

of the coefficients. On the other hand, Assumption 2.1-(i) also guarantees pathwise uniqueness

for the solutions of (2.1) and (2.2) by the Yamada-Watanabe result (see [27, Ch. 5.2, Prop. 2.13]

and [27, Ch. 5.3, Rem. 3.3], among others). Therefore, (2.1) and (2.2) have a unique strong

solution due to [27, Ch. 5.3, Cor. 3.23] for any x ∈ I1 and y ∈ I2. Also, it follows from (2.5)

that the diffusion processes Xx and Y y are regular in I1 and I2, respectively; that is, Xx (resp.,

Y y) hits a point ζ (resp., ζ ′) with positive probability, for any x and ζ in I1 (resp., y and ζ ′ in

I2). Hence the state spaces I1 and I2 cannot be decomposed into smaller sets from which Xx

and Y y could not exit (see, e.g., [40, Ch. V.7]). Finally, there exist continuous versions of Xx

and Y y and we shall always refer to those versions throughout this paper.

Assumption 2.1 implies the Yamada-Watanabe comparison criterion (see, e.g., [27, Ch. 5.2,

Prop. 2.18]); i.e.,

x, x′ ∈ I1 , x ≤ x′ =⇒ Xxt ≤ Xx′

t , P-a.s. ∀t ≥ 0. (2.6)

Moreover, repeating arguments as in the proof of [27, Ch. 5.2, Prop. 2.13] one also finds

xn → x0 in I1 as n→ ∞ =⇒ Xxnt

L1

−→ Xx0t =⇒ Xxn

tP

−→ Xx0t , ∀t ≥ 0; (2.7)

Analogously, for the unique solution of (2.2) one has

y, y′ ∈ I2 , y ≤ y′ =⇒ Y yt ≤ Y y′

t , P-a.s. ∀t ≥ 0; (2.8)

and

yn → y0 in I2 as n→ ∞ =⇒ Y ynt

L1

−→ Y y0t =⇒ Y yn

tP

−→ Y y0t , ∀t ≥ 0. (2.9)

The Optimal Boundary of an Irreversible Investment Problem 6

Standard estimates on the solution of SDEs with coefficients having sublinear growth imply that

(cf., e.g., [30, Ch. 2.5, Cor. 12])

E

[|Xx

t |q]≤ κ0,q(1 + |x|q)eκ1,qt, E

[|Y y

t |q]≤ θ0,q(1 + |y|q)eθ1,qt, t ≥ 0, (2.10)

for any q ≥ 0, and for some κi,q := κi,q(µ1, σ1) > 0 and θi,q := θi,q(µ2, σ2) > 0, i = 0, 1.

Within this setting we consider a firm that incurs investment costs and a running cost c(x, z)

depending on the state of economy x and the production capacity z. The firm’s total expected

cost of production associated to an investment strategy ν ∈ V is

Jx,y,z(ν) := E

[∫ ∞

0e−rtc(Xx

t , Zz,νt )dt+

∫ ∞

0e−rtY y

t dνt

], (2.11)

for any (x, y, z) ∈ I1 × I2 × R+. Here r is a positive discount factor and the cost function

c : I1 × R+ 7→ R

+ satisfies

Assumption 2.2.

(i) c ∈ C0(I1 × R+;R+), c(x, ·) ∈ C1(R+) for every x ∈ I1, and cz ∈ Cα(I1 × R

+;R) for

some α > 0 (that is, cz is α-Holder continuous).

(ii) c(x, ·) is convex for all x ∈ I1 and cz(·, z) is nonincreasing for every z ∈ R+.

(iii) c and cz satisfy a polynomial growth condition with respect to x; that is, there exist locally

bounded functions ηo, γo : R+ 7→ R

+, and a constant β ≥ 0 such that

|c(x, z)| + |cz(x, z)| ≤ ηo(z) + γo(z)|x|β .

Throughout this paper we also make the following standard assumption that guarantees in

particular finiteness for our problem (see Remark 2.4-(3) and Lemma 2.6 below)

Assumption 2.3. r > κ1,β ∨ θ1,1,

with κ1,q and θ1,q, q ≥ 0, as in (2.10) and with β of Assumption 2.2-(iii).

Remark 2.4. 1. Any function c of the spread |x− z| between capacity and demand in the form

c(x, z) = K0|x− z|δ , K0 ≥ 0, δ > 1, (2.12)

satisfies Assumption 2.2. We observe that (2.12) is a natural choice, e.g., in an energy market

framework where x represents the demand net of renewables (thus having stochastic nature) and

z the amount of conventional supply. Failing to meet the demand as well as an excess of supply

generate costs for the energy provider.

2. The second part of Assumption 2.2-(ii) captures the negative impact on marginal costs due

to an increase of demand. It is intuitive in (2.12) that an increase of z will produce a reduction

(increase) of costs which is more significant the more the demand is above (below) the supply.

3. It follows from (2.10), Assumption 2.2-(iii) and Assumption 2.3 that c and cz satisfy the

integrability conditions

The Optimal Boundary of an Irreversible Investment Problem 7

(a) E

[ ∫ ∞

0e−rtc(Xx

t , z)dt

]<∞, ∀(x, z) ∈ I1 × R

+;

(b) E

[ ∫ ∞

0e−rt|cz(X

xt , z)|dt

]<∞, ∀(x, z) ∈ I1 × R

+.

The firm’s manager aims at picking an irreversible investment policy ν∗ ∈ V (cf. (2.3))

that minimizes the total expected cost (2.11). Therefore, by denoting the state space O :=

I1×I2×R+, the firm’s manager is faced with the optimal irreversible investment problem with

value function

V (x, y, z) := infν∈V

Jx,y,z(ν), (x, y, z) ∈ O. (2.13)

Remark 2.5. The form of our cost functional (2.11) does not allow a reduction of the dimen-

sionality of problem (2.13) through an appropriate change of measure when Y is a discounted

exponential martingale (e.g., a geometric Brownian motion). That could have been possible in-

stead in the context of profit maximization problems with separable operating profit functions, as

the Cobb-Douglas one.

Notice that (2.10), Assumption 2.2-(ii) and Assumption 2.3 (cf. also Remark 2.4-(3)), to-

gether with the convexity of c(x, ·) and the affine nature of Zz,ν in the control variable lead to

the following

Lemma 2.6. The value function V (x, y, z) of (2.13) is finite for all (x, y, z) ∈ O and such that

z 7→ V (x, y, z) is convex.

Remark 2.7. If an optimal control ν∗ exists, then it must be Jx,y,z(ν∗) ≤ Jx,y,z(0) and hence

E

[∫ ∞

0e−r tc(Xx

t , z + ν∗t )dt

]< +∞. (2.14)

Therefore, there is no loss of generality if we restrict the set of admissible controls to those in

V which also fulfill (2.14).

Problem (2.13) is a degenerate, three-dimensional, convex singular stochastic control problem

of monotone follower type (see, e.g., [16], [26] and references therein). Moreover, if c is strictly

convex, then Jx,y,z(·) of (2.11) is strictly convex on V as well, and hence if a solution to (2.13)

exists, it must be unique. Existence of a solution ν∗ of convex (concave) singular stochastic

control problems is a well known result in the literature (see, e.g., [27], [28] or [39]) and it

usually relies on an application of (a suitable version of) Komlos’ Theorem.

Here we follow a different approach and in Section 5 we provide the optimal control ν∗ in

terms of the free-boundaries of a suitable family of optimal stopping problems that we start

studying in the next section.

The Optimal Boundary of an Irreversible Investment Problem 8

3 The Family of Associated Optimal Stopping Problems

In the literature on stochastic, irreversible investment problems (cf. [1], [11], [19], [29], [39],

among many others), or more generally on singular stochastic control problems of monotone

follower type (see, e.g., [3], [5], [16] and [26]), it is well known that a convex (concave) mono-

tone control problem may be associated to a suitable family of optimal stopping problems,

parametrized with respect to the state space of the controlled variable (see also [13], [18] and

[28] in the case of a bounded variation control problem, whose associated optimal stopping

problem is a Dynkin game).

We now introduce the family of optimal stopping problems that we expect to be associated

to the singular control problem (2.13). Set

T := τ ∈ [0,∞] F-stopping times,

and define

Ψx,y,z(τ) := E

[ ∫ τ

0e−rtcz(X

xt , z)dt − e−rτY y

τ

], τ ∈ T , (x, y) ∈ I1 × I2, z ∈ R

+. (3.1)

For any z ∈ R+ we consider the optimal stopping problem

v(x, y; z) := supτ∈T

Ψx,y,z(τ), (x, y) ∈ I1 × I2. (3.2)

Notice thatv(x, y; z), z ∈ R

+is a family of two-dimensional parameter-dependent optimal

stopping problems.

The basic formal connections one expects between the singular stochastic control problem

(2.13) and the optimal stopping problem (3.2) are the following (see, e.g., [1, Sec. 5]):

1. For fixed (x, y, z) ∈ O the first optimal stopping time τ∗ of problem (3.2) can be defined

in terms of the optimal control ν∗ of problem (2.13) by1

τ∗ = inft ≥ 0 : ν∗t > 0. (3.3)

2. The value function V of (2.13) is differentiable with respect to z and

Vz(x, y, z) = v(x, y; z), (x, y, z) ∈ O. (3.4)

Remark 3.1. The optimality of τ∗ in (3.3), the existence of Vz and the equality (3.4) may

be proved directly by suitably adapting to our setting the techniques employed in [1] or [26].

However we obtain these results as a byproduct of our verification theorem in Section 5.

In the rest of the present section and in the next one, we fix z ∈ R+ and we study the optimal

stopping problem (3.2). Denote its state space by Q := I1 ×I2. We introduce the following (cf.

[27, Ch. 1, Def. 4.8])

1 From the economic point of view, this means that a firm’s manager who aims at optimally (irreversibly)

investing may equivalently consider the problem of profitably exercising the investment option.

The Optimal Boundary of an Irreversible Investment Problem 9

Definition 3.2. A right-continuous stochastic process ξ := ξt, t ≥ 0 is of class (D) if the

family of random variables ξτ1τ<∞, τ ∈ T is uniformly integrable,

and we make the following technical

Assumption 3.3. The process e−rtY yt , t ≥ 0 is an (Ft)-supermartingale of class (D).

Remark 3.4. 1. The gain process e−rtY yt is of class (D) if, e.g., E[supt≥0 e

−rtY yt ] < ∞, a

standard technical assumption in the general theory of optimal stopping (see, e.g, [35, Ch. I]).

2. Assumptions 2.3 and 3.3 imply that limt→∞ e−rtY yt = 0 P-a.s. In fact, e−rtY y

t , t ≥ 0 is a

positive (Ft)-supermartingale with continuous paths (cf. also Assumption 2.1) and there always

exists Ξ := limt→∞ e−rtY yt ≥ 0 (cf. [27, Ch. 1, Problem 3.16]). Fatou’s Lemma gives

0 ≤ E[Ξ] = E[ limt→∞

e−rtY yt ] ≤ lim inft→∞E[e−rtY y

t ]

and, estimates in (2.10) and Assumption 2.3 imply limt→∞ E[e−rtY yt ] = 0, hence E[Ξ] = 0.

Since Ξ ≥ 0 P-a.s., then limt→∞ e−rtY yt = 0 P-a.s.

In light of Remark 3.4 from now on we will adopt the convention

e−rτY yτ 1τ=∞ := lim

t→∞e−rtY y

t = 0, a.s. (3.5)

Also we set

e−rτ |f(Xxτ , Y

yτ )|1τ=∞ := lim sup

t→∞e−rt|f(Xx

t , Yyt )|, a.s., (3.6)

for any Borel-measurable function f .

The next lemma will be useful in what follows.

Lemma 3.5. Under Assumptions 2.1, 2.3 and 3.3 it holds

E[e−rτY yτ ] = y + E

[∫ τ

0e−rt

(µ2(Y

yt )− rY y

t

)dt

], for τ ∈ T . (3.7)

Proof. The result holds for bounded stopping times τn := τ ∧ n, with τ ∈ T and n ∈ N,

by Ito’s formula and since the stochastic integral is a true martingale by Assumptions 2.1 and

2.3. Taking limits as n → ∞ and using Assumptions 2.1, 2.3, 3.3 and dominated convergence

one finds (3.7).

In the rest of this section we aim at characterizing v of (3.2).

Proposition 3.6. Under Assumptions 2.1, 2.2, 2.3 and 3.3 the following hold:

1. v is such that

−y ≤ v(x, y; z) ≤ C(z)(1 + |x|β + |y|), ∀(x, y) ∈ Q, (3.8)

for a constant C(z) > 0 depending on z.

The Optimal Boundary of an Irreversible Investment Problem 10

2. v( · , y; z) is nonincreasing for every y ∈ I2.

3. v(x, · ; z) is nonincreasing for every x ∈ I1.

Proof. 1. The lower bound follows by taking τ = 0 in (3.2). Assumptions 2.1, 2.2-(iii), 2.3,

3.3 and Lemma 3.5 guarantee the upper bound.

2. The fact that x 7→ cz(x, z) is nonincreasing (cf. Assumption 2.2-(ii)) and (2.6) imply

v(x2, y; z)− v(x1, y; z) ≤ supτ∈T

E

[ ∫ τ

0e−rt

(cz(X

x2t , z) − cz(X

x1t , z)

)dt]≤ 0, for x2 > x1.

3. It follows from (2.8) and arguments as in point 2.

Proposition 3.7. Under Assumptions 2.1, 2.2, 2.3 and 3.3 the value function v( · ; z) of the

optimal stopping problem (3.2) is continuous on Q.

Proof. Fix z ∈ R+ and let (xn, yn), n ∈ N ⊂ Q be a sequence converging to (x, y) ∈ Q.

Take ε > 0 and let τ ε := τ ε(x, y; z) be an ε-optimal stopping time for the optimal stopping

problem with value function v(x, y; z). Then we have

v(x, y; z) − v(xn, yn; z) ≤ ε+ E

[ ∫ τε

0e−rt

(cz(X

xt , z) − cz(X

xnt , z)

)dt− e−rτε(Y y

τε − Y ynτε )

].

(3.9)

Taking into account (2.7) and (2.9), Assumptions 2.2, 2.3 and 3.3, we can apply dominated

convergence (in its weak version requiring only convergence in measure; see, e.g., [8, Ch. 2, Th.

2.8.5]) to the right hand side of the inequality above and get

lim infn→∞

v(xn, yn; z) ≥ v(x, y; z) − ε. (3.10)

Similarly, taking ε-optimal stopping times τ εn := τ ε(xn, yn; z) for the optimal stopping pro-

blem with value function v(xn, yn; z), and using Lemma 3.5 we get

v(xn, yn; z) − v(x, y; z) ≤ ε+ E

[ ∫ τεn

0e−rt

(cz(X

xnt , z) − cz(X

xt , z)

)dt− e−rτεn

(Y ynτεn

− Y yτεn

)]

= ε+ E

[ ∫ τεn

0e−rt

(cz(X

xnt , z) − cz(X

xt , z)

)dt

]− (yn − y)

+ E

[∫ τεn

0e−rt

[r(Y ynt − Y y

t

)−

(µ2(Y

ynt )− µ2(Y

yt )

)]dt

](3.11)

≤ ε+ E

[ ∫ ∞

0e−rt

∣∣cz(Xxnt , z)− cz(X

xt , z)

∣∣dt]+ |y − yn|

+ C E

[ ∫ ∞

0e−rt

∣∣Y ynt − Y y

t

∣∣dt],

for some C > 0 and where we have used Lipschitz continuity of µ2 (cf. Assumption 2.1) in the

last step. Recalling now (2.7) and (2.9), (2.10), Assumptions 2.2 and 2.3, we can apply again

The Optimal Boundary of an Irreversible Investment Problem 11

dominated convergence in its weak version (cf. [8, Ch. 2, Th. 2.8.5]) to the right hand side of

the inequality above to obtain

lim supn→∞

v(xn, yn; z) ≤ v(x, y; z) + ε. (3.12)

Now (3.10) and (3.12) imply continuity of v( · , · ; z) by arbitrariness of ε > 0.

Remark 3.8. Arguments similar to those used in the proof of Proposition 3.7 above may also

be employed to show that (x, y, z) 7→ v(x, y; z) is continuous in O.

Since the state space Q = I1×I2 of the diffusion (Xxt , Y

yt ), t ≥ 0 may be unbounded, it is

convenient for studying the variational inequality associated to our optimal stopping problem,

to approximate problem (3.2) by a sequence of problems on bounded domains. Let Qn, n ∈ N

be a sequence of sets approximating Q, and we assume that

Qn is open, bounded and connected for every n ∈ N,

Qn ⊂ Q for every n ∈ N,

∂Qn ∈ C2+α for some α > 0 depending on n ∈ N,

Qn ⊂ Qn+1 for every n ∈ N,

limn→∞Qn :=⋃

n≥0Qn = Q.

(3.13)

Clearly it is always possible to find such a sequence of sets. The optimal stopping problem (3.2)

is then localized as follows. Given n ∈ N define the stopping time

τn = τn(x, y; z) := inft ≥ 0 | (Xxt , Y

yt ) /∈ Qn (3.14)

and notice that τ∞ = τ∞(x, y; z) := inft ≥ 0 | (Xxt , Y

yt ) /∈ Q = ∞ a.s., since we are assuming

that the boundaries of the diffusions Xx and Y y are natural, hence non attainable. Moreover,

from the last of (3.13) we obtain

τn ↑ τ∞ = ∞ P-a.s., as n→ ∞. (3.15)

With τn as in (3.14), we can define the approximating optimal stopping problem

vn(x, y; z) := supτ∈T

E

[ ∫ τn∧τ

0e−rtcz(X

xt , z)dt− e−r(τn∧τ)Y y

τn∧τ

], (x, y) ∈ Q, (3.16)

and prove the following

Proposition 3.9. Let Assumptions 2.1, 2.2, 2.3 and 3.3 hold. Then

1. vn( · ; z) ≤ vn+1( · ; z) ≤ v( · ; z) on Q for all n ∈ N.

2. vn(x, y; z) = −y for (x, y) ∈ Q \Qn and all n ∈ N (in particular for every (x, y) ∈ ∂Qn,

since Qn is open).

The Optimal Boundary of an Irreversible Investment Problem 12

3. vn(x, y; z) ↑ v(x, y; z) as n→ ∞ for every (x, y) ∈ Q.

4. If vn( · ; z), n ∈ N ⊂ C0(Q), then vn( · ; z) converges to v( · ; z) uniformly on all compact

subsets K ⊂⊂ Q.

Proof. 1. It follows from (3.15) and by comparison of (3.16) with (3.2).

2. This claim follows from the definition of τn and of vn (see (3.14) and (3.16), respectively).

3. For fixed (x, y) ∈ Q denote by τ ε := τ ε(x, y; z) an ε-optimal stopping time of v(x, y; z),

then

0 ≤ v(x, y; z) − vn(x, y; z)

≤ E

[∫ τε

τn∧τεe−rtcz(X

xt , z)dt−

(e−rτεY y

τε − e−rτnY yτn

)1τn<τε

]+ ε,

where the first inequality is due to 1 above. Now, the sequence of random variables Zn, n ∈ N

defined by

Zn :=

∫ τε

τn∧τεe−rtcz(X

xt , z)dt−

(e−rτεY y

τε − e−rτnY yτn

)1τn<τε

is uniformly integrable due to Assumptions 2.2, 2.3 and 3.3, and limn→∞Zn = 0 P-a.s., by

Remark 3.4-(2) and (3.15). Then 3 follows from Vitali’s convergence theorem and arbitrariness

of ε.

4. Since v( · ; z) ∈ C0(Q), the claim follows from 1 and 3 above and by Dini’s Lemma.

Remark 3.10. For each n ∈ N, the continuity of vn( · ; z) can be proved by its definition (3.16).

However, we will obtain it as a byproduct of the characterization of vn( · ; z) as the solution of a

suitable variational inequality.

Denote by L the second order elliptic differential operator associated to the two-dimensional

diffusion (Xt, Yt), t ≥ 0. Since X and Y are independent then L := LX + LY , with

(LXf) (x, y) :=1

2(σ1)

2(x)∂2

∂x2f(x, y) + µ1(x)

∂

∂xf(x, y),

(LY f) (x, y) :=1

2(σ2)

2(y)∂2

∂y2f(x, y) + µ2(y)

∂

∂yf(x, y),

for f ∈ C2b (Q). Fix n ∈ N and z ∈ R

+. From standard arguments we can formally associate the

function vn( · , · ; z)|Qn to the variational inequality (parametrized in z)

max(

L− r)u(x, y; z) + cz(x, z),−u(x, y; z) − y

= 0, (x, y) ∈ Qn, (3.17)

with boundary condition

u(x, y; z) = −y, (x, y) ∈ ∂Qn. (3.18)

Proposition 3.11. Under Assumptions 2.1, 2.2, 2.3 and 3.3, for each n ∈ N and z ∈ R+ there

exists a unique function un(· ; z) ∈W 2,p(Qn) for all 1 ≤ p <∞, satisfying (3.17) a.e. in Qn and

the boundary condition (3.18).

The Optimal Boundary of an Irreversible Investment Problem 13

Proof. Since µi, σi, i = 1, 2 are bounded and continuous on Qn, it suffices to apply [22,

Ch. I, Th. 3.2 and Th. 3.4].

Remark 3.12. Note that by well known Sobolev’s inclusions (see for instance [10, Ch. 9, Cor.

9.15]), the space W 2,p(Qn) with p ∈ (2,∞) can be continuously embedded into C1(Qn). Hence,

the boundary condition (3.18) is well-posed for functions in the class W 2,p(Qn), p ∈ (2,∞). In

the following we shall always refer to the unique C1 representative of elements of W 2,p(Qn).

The function un( · ; z) of Proposition 3.11 can be continuously extended outside Qn by setting

un(x, y; z) = −y, (x, y) ∈ Q \Qn. (3.19)

We denote such extension again by un with a slight abuse of notation.

Denote by Lqr(R+), q ∈ [1,∞), the Lq-spaces on R

+ with respect to the measure e−rsds. We

recall that X and Y are independent and make the following

Assumption 3.13. For every (x, y) ∈ I1 × I2 and t ≥ 0 the laws of Xxt and Y y

t have densities

p1(t, x, · ) and p2(t, y, · ), respectively. Moreover

1) (t, ζ, ξ) 7→ pi(t, ζ, ξ) is continuous on (0,∞)× Ii × Ii, i = 1, 2;

2) For any compact set K ⊂ I1 × I2 there exists q > 1 (possibly depending on K) such that

p1( · , x, · )p2( · , y, · ) ∈ L1r(R

+;Lq(K)), for all (x, y) ∈ K.

Remark 3.14. Assumption 3.13 is clearly satisfied in the benchmark case of X and Y given by

two independent geometric Brownian motions. The literature on the existence and smoothness

of densities for the probability laws of solutions of SDEs driven by Brownian motion is huge

and it mainly relies on PDEs’ and Malliavin Calculus’ techniques (see, e.g., [21] and [34] as

classical references on the topic). In general, the existence of a density for the law of a one-

dimensional diffusion is guaranteed under some very mild assumptions (see, e.g., the recent

paper [20]). Sufficient conditions on our (µi, σi), i = 1, 2, to obtain Gaussian bounds for the

transition densities and their first derivatives may be found for instance in [21, Ch. 1, Th. 11].

One can also refer to, e.g., [15] and references therein for more recent generalizations under

weaker assumptions.

Let us define the continuation and stopping regions of our approximating optimal stopping

problem (3.16) respectively by

Cnz := (x, y) ∈ Q | vn(x, y; z) > −y, An

z := (x, y) ∈ Q | vn(x, y; z) = −y. (3.20)

We provide now a verification theorem linking vn of (3.16) to un of Proposition 3.11.

Proposition 3.15. Let Assumptions 2.1, 2.2, 2.3, 3.3 and 3.13 hold and let n ∈ N. Then

vn( · ; z) = un( · ; z) over Qn. Moreover, the stopping time

τ∗n(x, y; z) := inft ≥ 0 | (Xx

t , Yyt ) /∈ Cn

z

(3.21)

is optimal for problem (3.16).

The Optimal Boundary of an Irreversible Investment Problem 14

Proof. Recall that un has been extended to Q in (3.19). If (x, y) ∈ Q \Qn, then the claim

clearly follows from Proposition 3.9-(2). Assume (x, y) ∈ Qn; since un ∈ W 2,p(Qn), by [23,

Ch. 7.6] we can find a sequenceu kn ( · ; z) , k ∈ N

⊂ C∞(Q) such that u k

n ( · ; z) → un( · ; z) in

W 2,p(Qn), p ∈ [1,+∞), as k → ∞. Moreover, since un is continuous and Qn is a compact, we

have u kn ( · ; z) → un( · ; z) uniformly on Qn (cf. [23, Ch. 7.2, Lemma 7.1]).

Dynkin’s formula yields for any bounded stopping time τ

ukn(x, y; z) = E

[e−r(τ∧τn)ukn(X

xτ∧τn , Y

yτ∧τn ; z)−

∫ τ∧τn

0e−rt(L− r)ukn(X

xt , Y

yt ; z) dt

]. (3.22)

Then by localization arguments and using (3.6), (3.22) actually holds for any τ ∈ T . We claim(and we will prove it later) that taking limits as k → ∞ in (3.22) leads to

un(x, y; z) = E

[e−r(τ∧τn)un(X

xτ∧τn

, Y yτ∧τn

; z)−

∫ τ∧τn

0

e−rt(L− r)un(Xxt , Y

yt ; z) dt

], ∀τ ∈ T . (3.23)

The right-hand side of (3.23) is well defined since Assumption 3.13 implies that the law of

(Xx, Y y) is absolutely continuous with respect to the Lebesgue measure and (L−r)un is defined

up to a Lebesgue null-measure set. We now use the variational inequality (3.17) in (3.23) to

obtain

un(x, y; z) ≥ E

[−e−r(τ∧τn)Y y

τ∧τn +

∫ τ∧τn

0e−rtcz(X

xt , z) dt

]. (3.24)

Hence, by arbitrariness of τ , one has un(x, y; z) ≥ vn(x, y; z).

To obtain the reverse inequality take

τ := inft ≥ 0 |un(X

xt , Y

yt ; z) = −Y y

t

(3.25)

in (3.23) and recall that un = −y on Q \ Qn, that un ∈ C0(Qn) (cf. Remark 3.12) and Qn is

bounded so that un is bounded in Qn as well. It follows that

e−r(τ∧τn)un(Xxτ∧τn , Y

yτ∧τn ; z) = e−r(τ∧τn)un(X

xτ∧τn , Y

yτ∧τn ; z)1τ∧τn<∞

=− e−r(τ∧τn)Y yτ∧τn1τ∧τn<∞ = −e−r(τ∧τn)Y y

τ∧τn P-a.s. (3.26)

by (3.5) and (3.6). Moreover, by (3.17), we have (LX − r)un = −cz on the set(x, y) ∈

Qn |un(x, y; z) > −y. Hence (3.23) and (3.26) give

un(x, y; z) = E

[−e−r(τ∧τn)Y y

τ∧τn +

∫ τ∧τn

0e−rtcz(X

xt , z) dt

]≤ vn(x, y; z). (3.27)

Therefore, we conclude that un = vn on Q, and that the stopping time τ defined in (3.25) is

optimal for problem (3.16) and coincides with the stopping time τ∗n(x, y; z) defined in (3.21).

Now, to complete the proof we only need to show that (3.23) follows from (3.22) as k → ∞.

In fact, the term on the left-hand side of (3.22) converges pointwisely and the first term in the

expectation on the right-hand side converges by uniform convergence. To check convergence of

The Optimal Boundary of an Irreversible Investment Problem 15

the integral term in the expectation on the right-hand side we take qn > 0 as in Assumption

3.13-(2), pn such that 1pn

+ 1qn

= 1 and for simplicity denote q := qn and p := pn. Then, by

Holder’s inequality we have

∣∣∣∣E[∫ τ∧τn

0e−rt(L − r)(u k

n − un)(Xxt , Y

yt ; z) dt

] ∣∣∣∣

≤

∫ ∞

0e−rt

(∫

Qn

∣∣(L− r)(u kn − un)(ξ, ζ; z)

∣∣ p1(t, x, ξ)p2(t, y, ζ)dξ dζ)dt (3.28)

≤ CM1,M2,r,n

∥∥u kn − un

∥∥W 2,p(Qn)

where last inequality follows by Assumptions 2.1-(i) and 3.13-(2) with CM1,M2,r,n > 0 depending

on Qn, r and Mi := supQn|µi|+ |σi| , i = 1, 2. Now, the right-hand side of (3.28) vanishes as

k → ∞ by definition of ukn.

Lemma 3.16. One has

(L− r)vn(x, y) = ry − µ2(y), for a.e. (x, y) ∈ Anz ∩Qn. (3.29)

Proof. Recall that vn ≡ un and that un( · ; z) ∈ W 2,p(Qn) (cf.(3.19), Proposition 3.15,

Proposition 3.11 and (3.19), respectively). Set vn(x, y; z) := vn(x, y; z) + y, hence vn ∈ C1(Qn)

by Sobolev’s embedding (see for instance [10, Ch. 9, Cor. 9.15]) and proving (3.29) amounts to

showing that (L− r)vn = 0 a.e. on Anz ∩Qn. Since vn = 0 over An

z , it must also be ∇vn = 0 over

Anz ∩Qn. To complete the proof it thus remains to show that the Hessian matrix D2vn is zero

a.e. over Anz ∩Qn. This follows by [17, Cor. 1-(i), p. 84]2 with f therein defined by f := ∇vn.

Proposition 3.17. For every (x, y) ∈ Q the following representation holds

vn(x, y; z) = E

[∫ τn

0e−rt

(cz(X

xt , z)1(Xx

t ,Yyt )∈Cn

z −(rY y

t −µ2(Yyt ))1(Xx

t ,Yyt )∈An

z

)dt− e−rτnY y

τn

].

(3.30)

Proof. Taking τ = ∞ in (3.23) and considering (3.26) and Proposition 3.15, we get

vn(x, y; z) = E

[−e−rτnY y

τn−

∫ τn

0e−rt(L− r)vn(X

xt , Y

yt ; z) dt

]. (3.31)

It follows from Propositions 3.11, 3.15 and from Lemma 3.16 that

(L− r)vn(x, y; z) = cz(x, z)1(x,y)∈Cnz − (ry − µ2(y))1(x,y)∈An

z , for a.e. (x, y) ∈ Qn, (3.32)

and we have the claim by using (3.31) and Assumption 3.13 in (3.32).

2It is worth noting that [17, Cor. 1-(i), p. 84] requires f to be Lipschitz continuous, which is not guaranteed

for us. However Lipschitz continuity is only needed there to have existence a.e. of the gradient ∇f , which we have

due to [17, Th. 1, p. 235] since ∇vn ∈ W 1,p(Qn).

The Optimal Boundary of an Irreversible Investment Problem 16

We now aim at proving a probabilistic representation of v similar to (3.30). The idea is

to pass (3.30) to the limit as n ↑ ∞ and use Proposition 3.9. For that we first define the

continuation and stopping regions of problem (3.2) as

Cz := (x, y) ∈ Q | v(x, y; z) > −y, Az := (x, y) ∈ Q | v(x, y; z) = −y. (3.33)

It is worth recalling that (3.1) and standard arguments based on exit times from small subsets

of Q give the following inclusion

Az ⊂ L−z :=

(x, y) ∈ Q | cz(x, z) ≤ µ2(y)− ry

. (3.34)

We observe that since vn ≤ v and vn, n ∈ N is an increasing sequence then

Cnz ⊂ Cn+1

z ⊂ Cz, Anz ⊃ An+1

z ⊃ Az, ∀n ∈ N. (3.35)

On the other hand, the pointwise convergence vn ↑ v (cf. Proposition 3.9) implies that if (x0, y0) ∈

Cz, then v(x0, y0)+y0 ≥ ε0 for some ε0 > 0 and vn(x0, y0)+y0 ≥ ε0/2 for all n ≥ n0 and suitable

n0 ∈ N. Hence we have

limn→∞

Cnz :=

⋃

n≥0

Cnz = Cz, lim

n→∞An

z :=⋂

n≥0

Anz = Az (3.36)

and the following representation result.

Theorem 3.18. Under Assumptions 2.1, 2.2, 2.3, 3.3 and 3.13 the following representation

holds for every (x, y) ∈ Q:

v(x, y; z) = E

[ ∫ ∞

0e−rt

(cz(X

xt , z)1(Xx

t ,Yyt )∈Cz − (rY y

t − µ2(Yyt ))1(Xx

t ,Yyt )∈Az

)dt

]. (3.37)

Proof. We study (3.30) in the limit as n ↑ ∞. Observe that:

1. The left-hand side of (3.30) converges pointwisely to v(x, y; z) by Proposition 3.9-(3);

2. e−rτnY yτn , n ∈ N is a family of random variables uniformly integrable and converging a.s.

to 0, due to (3.15) and to Assumptions 2.3 and 3.3 (see also the discussion in Remark 3.4-(2)).

Hence limn→∞ E [e−rτnY yτn ] = 0, by Vitali’s convergence Theorem;

3. From (3.35), one has∣∣∣∣E

[ ∫ τn

0e−rtcz(X

xt , z)1(Xx

t ,Yyt )∈Cndt−

∫ ∞

0e−rtcz(X

xt , z)1(Xx

t ,Yyt )∈Cdt

]∣∣∣∣ (3.38)

≤ E

[ ∫ ∞

0e−rt|cz(X

xt , z)|1(Xx

t ,Yyt )∈C \ Cndt

]+ E

[ ∫ ∞

τn

e−rt|cz(Xxt , z)|1(Xx

t ,Yyt )∈Cdt

].

The first term in the right-hand side of (3.38) converges to zero as n → ∞ by dominated

convergence and (3.36) (cf. Assumptions 2.2-(iii), 2.3 and Remark 2.4-(3)). Similarly, dominated

convergence and (3.15) give

limn→∞

E

[ ∫ ∞

τn

e−rt|cz(Xxt , z)|1(Xx

t ,Yyt )∈Cdt

]= 0.

The Optimal Boundary of an Irreversible Investment Problem 17

4. From (3.36) it follows that for a.e. (t, ω) ∈ R+ × Ω

limn→∞

1[0,τn](t)e−rt

[rY y

t − µ2(Yyt )

]1(Xx

t ,Yyt )∈An

z = e−rt

[rY y

t − µ2(Yyt )

]1(Xx

t ,Yyt )∈Az.

Moreover, due to Lipschitz-continuity of µ2 (cf. Assumption 2.1),

∣∣∣e−rt[rY y

t − µ2(Yyt )

]1(Xx

t ,Yyt )∈An

z

∣∣∣ ≤ e−rt∣∣∣rY y

t − µ2(Yyt )

∣∣∣ ≤ e−rtC0(1 + Y yt ),

for some C0 > 0 depending on y and r. The last expression of the inequality above is integrable

in R+ × Ω by (2.10) and by Assumption 2.3. Hence dominated convergence and (3.15) yield

limn→∞

E

[∫ τn

0e−rt

[rY y

t − µ2(Yyt )

]1(Xx

t ,Yyt )∈An

z dt

]= E

[∫ ∞

0e−rt

[rY y

t − µ2(Yyt )

]1(Xx

t ,Yyt )∈Azdt

].

Now taking n→ ∞ in (3.30) and using 1-4 above, (3.37) follows.

Set

H(x, y; z) := cz(x, z)1(x,y)∈Cz − (ry − µ2(y))1(x,y)∈Az (3.39)

so that (3.37) may be written as

v(x, y; z) = E

[ ∫ ∞

0e−rtH(Xx

t , Yyt ; z)dt

]. (3.40)

Due to (3.8) and Assumption 2.3, the strong Markov property and standard arguments based

on conditional expectations applied to the representation formula (3.40) allow to verify that

e−rtv(Xx

t , Yyt ; z) +

∫ t

0e−rsH(Xx

s , Yys ; z)ds, t ≥ 0

is an (Ft)-martingale, (3.41)

for all (x, y) ∈ Q.

By similar methods one can check that

∣∣e−rτv(Xxτ , Y

yτ ; z)

∣∣ ≤ E

[ ∫ ∞

0e−rt

∣∣H(Xxt , Y

yt ; z)

∣∣ dt∣∣∣Fτ

], τ ∈ T , (3.42)

and hence

the familye−rτv(Xx

τ , Yyτ ; z) , τ ∈ T

is uniformly integrable. (3.43)

Now, recalling (3.34) and according to standard theory of optimal stopping (cf., e.g., [35, Th.

2.4]), the martingale property (3.41) gives

Theorem 3.19. Fix (x, y) ∈ Q. Under Assumptions 2.1, 2.2, 2.3, 3.3 and 3.13, the process

S :=

e−rtv(Xx

t , Yyt ; z) +

∫ t

0e−rscz(X

xs , z)ds , t ≥ 0

(3.44)

The Optimal Boundary of an Irreversible Investment Problem 18

is an (Ft)-supermartingale and

E

[e−rτv(Xx

τ , Yyτ ; z) +

∫ τ

0e−rscz(X

xs , z)ds

]≤ v(x, y; z) , ∀τ ∈ T . (3.45)

Moreover, the stopping time

τ∗ = τ∗(x, y; z) := inf t ≥ 0 | v(Xxt , Y

yt ; z) = −Y y

t (3.46)

is optimal for problem (3.2) and the processSt∧τ∗ , t ≥ 0

is an (Ft)-martingale.

Proof. The supermartingale property (3.44) easily follows from (3.41) and (3.34). Similarly,

(3.45) is true for any σn := τ ∧ n with τ ∈ T and n ∈ N, i.e. (cf. (3.44))

E[Sσn ] ≤ S0. (3.47)

Then (3.45) is obtained by taking limits as n→ ∞ and by using dominated convergence, (3.43)

and the fact that Sσn → Sτ P-a.s. by Proposition 3.7 and continuity of paths.

For the optimality of τ∗ notice that (3.47) holds with equality if σn = τ∗ ∧ n and, moreover,(v(Xx

τ∗ , Yyτ∗ ; z) + Y y

τ∗

)1τ∗≤n = 0 P-a.s. Hence one has

v(x, y; z) = E

[∫ τ∗∧n

0e−rtcz(X

xt , z)dt− 1τ∗≤ne

−rτ∗Y yτ∗ + 1τ∗>ne

−rnv(Xxn , Y

yn ; z)

]. (3.48)

Taking limits as n→ ∞ and using Assumptions 2.2, 2.3, 3.3, Proposition 3.6-(1), and dominated

convergence one obtains

v(x, y; z) = E

[ ∫ τ∗

0e−rtcz(X

xt , z)dt − e−rτ∗Y y

τ∗

], (3.49)

hence optimality of τ∗. The martingale property ofSt∧τ∗ , t > 0

easily follows from the results

above.

4 Characterization of the Optimal Boundary

In this section we will provide a characterization of the optimal boundaries of the family of

optimal stopping problems (3.2). For that we define

y∗(x; z) := infy ∈ I2 | v(x, y; z) > −y, (x, z) ∈ I1 × R+, (4.1)

with the convention inf ∅ = y. Notice that under this convention y∗( · ; z) takes values in I2.

We will show that under suitable conditions y∗( · ; z) splits I1 × I2 into Cz and Az (cf. (3.33)).

Moreover, we will characterize y∗( · ; z) as the unique continuous solution of a nonlinear integral

equation of Fredholm type.

The Optimal Boundary of an Irreversible Investment Problem 19

Remark 4.1. Integral equations for the optimal boundaries of one-dimensional optimal stopping

problems on a finite time-horizon are often obtained by an application of the so-called local

time space calculus (cf. [36]). In order to do so in our case we should prove that the process

y∗(Xxt ; z), t ≥ 0 is a semimartingale for each given z ∈ R

+ as required in [36, Th. 2.1]. That

seems an extremely hard task and we will follow a different approach mainly based on the results

of Section 3 and probabilistic techniques.

We now make the following

Assumption 4.2. Assumptions 2.1, 2.2, 2.3, 3.3 and 3.13 hold. Moreover, the map y 7→

ry − µ2(y) is strictly increasing.

Proposition 4.3. Under Assumption 4.2 one has (cf. (3.33))

Cz = (x, y) ∈ Q | y > y∗(x; z), Az = (x, y) ∈ Q | y ≤ y∗(x; z). (4.2)

Proof. It suffices to show that y 7→ v(x, y; z) + y is nondecreasing for each x ∈ I1, z ∈ R+.

Set u := v+ y, take y1 and y2 in I2 such that y2 > y1 and set τ1 := inft ≥ 0 | (Xxt , Y

y1t ) /∈ Cz,

which is optimal for v(x, y1; z). From Lemma 3.5 and the superharmonic characterization of

Theorem 3.19 we obtain

v(x, y2; z)− v(x, y1; z) ≥ E

[e−rτ1

(v(Xx

τ1, Y y2

τ1; z) − v(Xx

τ1, Y y1

τ1; z)

)]

+ E

[ ∫ τ1

0e−rt

(r(Y y2t − Y y1

t

)−

(µ2(Y

y2t )− µ2(Y

y1t )

))dt

](4.3)

≥ E

[e−rτ1

(v(Xx

τ1, Y y2

τ1; z) − v(Xx

τ1, Y y1

τ1; z)

) ],

where the last inequality follows by (2.8) and Assumption 4.2. Note that the last expression in

(4.3) is well defined thanks to Assumption 3.3 and (3.43). Moreover, since v ≥ 0 it holds

E

[e−rτ1

(v(Xx

τ1, Y y2

τ1; z)− v(Xx

τ1, Y y1

τ1; z)

) ]≥ −E

[e−rτ1 v(Xx

τ1, Y y1

τ1, z)

]. (4.4)

By Assumption 2.3, Proposition 3.6-(1) and since 1τ1≤ne−rτ1 v(Xx

τ1, Y y1

τ1 ; z) = 0 P-a.s., Fatou’s

Lemma gives

E

[e−rτ1 v(Xx

τ1, Y y1

τ1; z)

]= E

[lim infn→∞

e−r(τ1∧n)v(Xxτ1∧n, Y

y1τ1∧n; z)

]

≤ lim infn→∞

E[e−rnv(Xx

n , Yy1n ; z)1τ1>n

]= 0 (4.5)

Now (4.3), (4.4) and (4.5) imply that y 7→ v(x, y; z) is increasing and therefore (4.2) holds.

Notice that (3.37) and (4.2) imply

v(x, y; z) = E

[ ∫ ∞

0e−rt

(cz(X

xt , z)1Y y

t >y∗(Xxt ;z)

− (rY yt − µ2(Y

yt ))1Y y

t ≤y∗(Xxt ;z)

)dt

]. (4.6)

The Optimal Boundary of an Irreversible Investment Problem 20

Under Assumption 3.13, (4.6) can also be expressed in a purely analytical way as

v(x, y; z) =

∫ ∞

0e−rt

[ ∫ x

x

p1(t, x, ξ)cz(ξ, z)

(∫ y

y∗(ξ;z)p2(t, y, η)dη

)dξ

]dt (4.7)

−

∫ ∞

0e−rt

[ ∫ x

x

p1(t, x, ξ)

(∫ y∗(ξ;z)

y

(rη − µ2(η))p2(t, y, η)dη

)dξ

]dt,

for any (x, y, z) ∈ O.

Proposition 4.4. Under Assumption 4.2 one has

1. the function y∗( · ; z) is nondecreasing and right-continuous for any z ∈ R+;

2. the function y∗(x; · ) is nonincreasing and left-continuous for any x ∈ I1;

Proof. Claims 1 and 2 follow by adapting arguments from the proof of [25, Prop. 2.2] and

by using our Proposition 3.6-(2)-(3), and Proposition 3.7.

It follows from Propositions 4.3 and 4.4-(1) that the regions Cz and Az are connected for

every z ∈ R+, and the optimal stopping time τ∗(x, y; z) defined in (3.46) can be written as

τ∗(x, y; z) = inft ≥ 0 |Y y

t ≤ y∗(Xxt ; z)

. (4.8)

Thanks to the representation (4.6) or (4.7), under the following further assumptions we can

prove the C1-regularity of the function v.

Assumption 4.5. The functions p1(t, ·, ξ) and p2(t, ·, η) are differentiable for each (t, ξ) ∈

R+ × I1 and each (t, η) ∈ R

+ × I2, respectively. Moreover, denoting by p′i, i = 1, 2 the partial

derivative of pi with respect to the second variable, it holds

1) x 7→ p′1(t, x, ξ) is continuous in I1 for all (t, ξ) ∈ R+ ×I1 and, for any (x, y, z) ∈ O, there

exists δ > 0 such that supζ∈[x−δ,x+δ]

∣∣p′1(t, ζ, ξ)∣∣ ≤ ψ1(t, ξ; δ) for some ψ1 such that

(t, ξ, η) 7→ e−rtψ1(t, ξ; δ)p2(t, y, η)(cz(ξ, z) + η) is in L1(R+ × I1 × I2); (4.9)

2) y 7→ p′2(t, y, η) is continuous in I2 for all (t, η) ∈ R+ ×I2 and, for any (x, y, z) ∈ O, there

exists δ > 0 such that supζ∈[y−δ,y+δ]

∣∣p′2(t, ζ, η)∣∣ ≤ ψ2(t, η; δ) for some ψ2 such that

(t, ξ, η) 7→ e−rtψ2(t, η; δ)p1(t, x, ξ)(cz(ξ, z) + η) is in L1(R+ × I1 × I2). (4.10)

Proposition 4.6. Under Assumptions 4.2 and 4.5 one has v( · ; z) ∈ C1(Q) for every z ∈ R+.

Proof. The proof follows by (4.7), by Assumption 4.5 and standard dominated convergence

arguments.

Proposition 4.6 above states in particular the so-called smooth-fit condition across the free-

boundary, i.e. the continuity of vx( · ; z) and vy( · ; z) at ∂Az. With the aim of characterizing the

boundary y∗( · ; z) as unique continuous solution of a (parametric) integral equation we make

the following additional

The Optimal Boundary of an Irreversible Investment Problem 21

Assumption 4.7. The drift coefficient µ2 is continuously differentiable in I2 and ∂µ2

∂y< r.

Moreover, (µ2, σ2) ∈ C1+δ(I2), for some δ > 0.

Proposition 4.8. Under Assumptions 4.2, 4.5 and 4.7, the function y∗( · ; z) : I1 → I2 is

continuous.

Proof. We know that the function y∗( · ; z) is nondecreasing and right-continuous by Propo-

sition 4.4-(1). Hence it suffices to show that it is also left-continuous. Arguing by contradiction,

we assume that there exists x0 ∈ I1 such that y∗(x0−; z) := limx↑x0 y∗(x; z) < y∗(x0; z). Then,

there also exist y0 ∈ I2 and ε > 0 such that

Σz := (x0 − ε, x0)× (y0 − ε, y0 + ε) ⊂ Cz, x0 × (y0 − ε, y0 + ε) ⊂ Az.

Notice that, by standard arguments on free-boundary problems and optimal stopping (cf. for

instance [35, Ch. 3, Sec. 7]), one has that v( · ; z) ∈ C2(Cz) and solves

1

2σ21(x)vxx(x, y; z) = −µ1(x)vx(x, y; z) − (LY − r)v(x, y; z) − cz(x, z), (x, y) ∈ Cz. (4.11)

On the other hand, since (µ2, σ2) ∈ C1+δ(I2), regularity results on uniformly elliptic partial

differential equations (cf. for instance [23, Ch. 6, Th. 6.17]) imply that one actually has vy( · ; z) ∈

C2+δ(Cz). Hence we can differentiate (4.11) with respect to y to find

1

2σ21(x)(vy)xx(x, y; z) = −µ1(x)(vy)x(x, y; z) − (R− r)vy(x, y; z), (x, y) ∈ Cz, (4.12)

where

(Rf)(x, y) :=1

2σ22(y)fyy(x, y) +

[∂σ22∂y

(y) + µ2(y)]fy(x, y) +

∂µ2∂y

(y)f(x, y), f ∈ C2b (Q).

Take now y1, y2 ∈ (y0 − ε, y0 + ε) with y1 < y2 and set

Fφ(x; y1, y2, z) := −

∫ y2

y1

vxx(x, y; z)φ′(y)dy, x ∈ (x0 − ε, x0), (4.13)

where φ is real-valued, arbitrarily chosen and such that

φ ∈ C∞c (y1, y2), φ ≥ 0,

∫ y2

y1

φ(y)dy > 0.

From now on we will write Fφ(x) instead of Fφ(x; y1, y2, z) to simplify the notation. Multiply

both sides of (4.12) by 2φ(y)/σ21(x) and integrate by parts with respect to y ∈ (y1, y2); it follows

Fφ(x) = −

∫ y2

y1

1

σ21(x)

[µ1(x)vxy(x, y; z) + (R− r)vy(x, y; z)

]φ(y)dy (4.14)

=µ1(x)

σ21(x)

∫ y2

y1

vx(x, y; z)φ′(y)dy +

1

σ21(x)

∫ y2

y1

v(x, y; z)∂

∂y(R− r)∗φ(y)dy,

The Optimal Boundary of an Irreversible Investment Problem 22

for every x ∈ (x0 − ε, x0), with (R − r)∗ denoting the adjoint of (R − r). Now, recalling

Proposition 4.6 and the definition of Cz and Az one also has

v(x0, y; z) = −y, ∀y ∈ [y1, y2],

vx(x0, y; z) = 0, ∀y ∈ [y1, y2],

vy(x0, y; z) = −1, ∀y ∈ [y1, y2],

(4.15)

and thus, taking limits in (4.14), one obtains

limx↑x0

Fφ(x) = −1

σ21(x0)

∫ y2

y1

y∂

∂y(R− r)∗φ(y)dy =

1

σ21(x0)

∫ y2

y1

[(R− r)1]φ(y)dy

=1

σ21(x0)

∫ y2

y1

( ∂

∂yµ2(y)− r

)φ(y)dy < 0, (4.16)

where the last inequality follows from Assumption 4.7. Since Fφ is clearly continuous in (x0 −

ε, x0), we see from (4.16) that it must be Fφ < 0 in a left neighborhood of x0 and, without any

loss of generality, we assume that Fφ < 0 in (x0 − ε, x0). Recalling (4.13), we have for each

δ ∈ (0, ε)

0 >

∫ x0

x0−δ

Fφ(x)dx = −

∫ x0

x0−δ

∫ y2

y1

vxx(x, y; z)φ′(y)dy dx

= −

∫ y2

y1

[vx(x0, y; z)− vx(x0 − δ, y; z)]φ′(y)dy

=

∫ y2

y1

vx(x0 − δ, y; z)φ′(y)dy = −

∫ y2

y1

vxy(x0 − δ, y; z)φ(y)dy,

by (4.15) and Fubini-Tonelli’s theorem. This implies that vxy( · ; z) > 0 in Σz by arbitrariness

of φ and δ and hence the function x 7→ vy(x, y; z) is strictly increasing in (x0 − ε, x0) for any

y ∈ [y1, y2]. It then follows from the last of (4.15)

vy( · ; z) < −1 in Σz ⊂ Cz. (4.17)

On the other hand, vy( · ; z) solves (4.12) subject to the boundary condition vy( · ; z) = −1 on

∂Cz by Proposition 4.6. Therefore it admits the standard Feynman-Kac representation (see,

e.g., [27, Ch. 5, Sec. 7.B])

vy(x, y; z) = E

[− e

∫ τCz0

(∂∂y

µ2(Yyt )−r

)dt], (4.18)

where τCz := inft ≥ 0 | (Xxt , Y

yt ) /∈ Cz, and with Y y solving

dY y

t =[∂σ2

2∂y

(Y yt ) + µ2(Y

yt )

]dt+ σ2(Y

yt )dW

2t , t > 0,

Y y0 = y.

Since r > ∂µ2

∂yby Assumption 4.7, (4.18) implies vy( · ; z) > −1 in Cz, contradicting (4.17) and

concluding the proof.

The Optimal Boundary of an Irreversible Investment Problem 23

In order to find an upper bound for y∗( · ; z) we now denote

F (x, y; z) := cz(x, z) − µ2(y) + ry, (x, y) ∈ Q, (4.19)

and define

ϑ(x; z) := infy ∈ I2 |F (x, y; z) > 0 ∈ I2, x ∈ I1, (4.20)

with the convention inf ∅ = y. Then by Proposition 4.3 and by (3.34), we have

y∗( · ; z) ≤ ϑ( · ; z). (4.21)

Lemma 4.9. Under Assumption 4.2 and 4.7, the function ϑ( · ; z) is nondecreasing and contin-

uous. Moreover, if ϑ(x; z) ∈ I2 then ϑ(x; z) is the unique solution to the equation F (x, ·; z) = 0

in I2. Finally one has(x, y) ∈ Q | cz(x, z) − µ2(y) + ry < 0

= (x, y) ∈ Q | y < ϑ(x; z). (4.22)

Proof. Since x 7→ F (x, y; z) is nonincreasing (cf. Assumption 2.2-(ii)) and y 7→ F (x, y; z)

is increasing by Assumption 4.7 and (x, y) 7→ F (x, y; z) it is not hard to see that ϑ(·; z) is

nondecreasing and right-continuous.

The definition of ϑ(·; z) and the continuity of F guarantee that if ϑ(x; z) ∈ I2 then ϑ(x; z)

solves F (x, ·; z) = 0 in I2. Assumption 4.7 then implies that ϑ(x; z) is actually the unique

solution of such equation.

Let us now show that ϑ( · ; z) is continuous. Take x0 such that ϑ(x0; z) > y and assume that

ϑ(x0−; z) < ϑ(x0; z). Take a sequence xn , n ∈ N ⊂ I1 increasing and such that xn ↑ x0. One

has F (xn, ϑ(xn; z); z) ≥ 0 for all n ∈ N and hence in the limit one finds F (x0, ϑ(x0−; z); z) ≥

0 ≥ F (x0, ϑ(x0; z); z) which implies ϑ(x0−; z) ≥ ϑ(x0; z) since y 7→ F (x, y; z) is increasing.

Clearly (4.22) follows from the previous properties.

Consider now the class of functions

Mz := f : I1 → I2, continuous, nondecreasing and dominated from above by ϑ( · ; z),

and define

Df := x ∈ I1 | f(x) ∈ I2, f ∈ Mz.

ClearlyMz is nonempty as ϑ( · ; z) ∈ Mz by Lemma 4.9, and Df is an open sub-interval (possibly

empty) of I1. We set

xf := infx ∈ I1 | f(x) > y, xf := supx ∈ I1 | f(x) < y, (4.23)

with the conventions inf ∅ = x, sup ∅ = x. Notice that by monotonicity of any arbitrary f ∈ Mz

we have f ≡ y on (x, xf ) (if the latter is nonempty) and, analogously, f ≡ y on (xf , x) (if the

latter is nonempty). Given a function y( · ; z) ∈ Mz, we set

H(x, y; z) := cz(x, z)1y>y(x;z) −(ry − µ2(y)

)1y≤y(x;z) (4.24)

The Optimal Boundary of an Irreversible Investment Problem 24

and define

w(x, y; z) := E

[∫ ∞

0e−rtH(Xx

t , Yyt ; z)dt

]. (4.25)

Notice that

∣∣w(x, y; z)∣∣ ≤ C(z)

(1 + |x|β + |y|

), for (x, y) ∈ Q, (4.26)

by Assumptions 2.1, 2.2, 2.3 (cf. also (3.8)). Moreover, as in (3.41) and (4.2) one can verify that

e−rtw(Xx

t , Yyt ; z) +

∫ t

0e−rsH(Xx

s , Yys ; z)ds, t ≥ 0

is an (Ft)-martingale (4.27)

and that

the familye−rτw(Xx

τ , Yyτ ; z) , τ ∈ T

is uniformly integrable. (4.28)

To simplify notation from now on we set

x := xy(·;z), x := xy(·;z), Dz := Dy(·;z) (4.29)

and

x∗ := xy∗(·;z), x∗ := xy∗(·;z), D∗z := Dy∗(·;z). (4.30)

We can now state the main result of this section. We use arguments inspired by [35, Sec. 25]

and references therein.

Theorem 4.10. Let Assumptions 4.2, 4.5 and 4.7 hold. Assume that Cz 6= ∅ and Az 6= ∅. Then

y∗( · ; z) is the unique nontrivial solution within the class Mz of the equation

−y(x; z) =

∫ ∞

0e−rt

[ ∫ x

x

p1(t, x, ξ)cz(ξ, z)

(∫ y

y(ξ;z)p2(t, y(x; z), η)dη

)dξ

]dt (4.31)

−

∫ ∞

0e−rt

[ ∫ x

x

p1(t, x, ξ)

(∫ y(ξ;z)

y

(rη − µ2(η))p2(t, y(x; z), η)dη

)dξ

]dt;

that is, y∗( · ; z) is the unique function y( · ; z) ∈ Mz with Dy(·;z) 6= ∅ and such that (4.31) holds

for each x ∈ Dy(·;z).

Proof. Existence. First of all we observe that y∗( · ; z) ∈ Mz by Propositions 4.4, 4.8 and

(4.21). The fact that y∗( · ; z) solves (4.31) for each x ∈ D∗z follows by evaluating both sides of

(4.6) at points of the boundary (x, y∗(x; z)) ∈ ∂Az, which yields

− y∗(x; z) =

∫ ∞

0e−rt

E

[cz(X

xt , z)1Y

y∗(x;z)t >y∗(Xx

t ;z)

]dt (4.32)

−

∫ ∞

0e−rt

E

[(rY

y∗(x;z)t − µ2(Y

y∗(x;z)t ))1

Yy∗(x;z)t ≤y∗(Xx

t ;z)

]dt.

The Optimal Boundary of an Irreversible Investment Problem 25

From (4.32) and by Assumption 3.13, we see that y∗( · ; z) solves (4.31).

Uniqueness. Let y( · ; z) ∈ Mz be a nontrivial solution of (4.31) and recall (4.29) and (4.30).

We need to show that y( · ; z) ≡ y∗( · ; z).

Step 1. Here we show that y( · ; z) ≥ y∗( · ; z).

Case (i): D∗z∩Dz 6= ∅. Assume by contradiction that y(x; z) < y∗(x; z) for some x ∈ D∗

z∩Dz,

take y < y(x; z) and set σ = σ(x, y, z) := inft ≥ 0 |Y y

t ≥ y∗(Xxt ; z)

.

Then from (3.41) and (4.27) it follows (up to localization arguments as in the proofs of

Theorem 3.19 and Lemma A.1) that

E[e−rσv(Xx

σ , Yyσ ; z)

]= v(x, y; z) + E

[∫ σ

0e−rt

(rY y

t − µ2(Yyt )

)dt

], (4.33)

E[e−rσw(Xx

σ , Yyσ ; z)

]= w(x, y; z) − E

[∫ σ

0e−rtH(Xx

t , Yyt ; z) dt

]. (4.34)

Lemma A.1 in Appendix A ensures that v ≥ w everywhere and that w(x, y; z) = v(x, y; z) = −y

since y < y(x; c) < y∗(x; z) (cf. (A-3)). Then subtracting (4.34) from (4.33) one has

0 ≤ E

[∫ σ

0e−rt

[(rY y

t − µ2(Yyt )

)+ H(Xx

t , Yyt ; z)

]dt

]

= E

[∫ σ

0e−rt

[cz(X

xt , z) −

(µ2(Y

yt )− rY y

t

)]1y(Xx

t ;z)<Yyt <y∗(Xx

t ;z)dt

]. (4.35)

Notice that the continuity of trajectories of (Xx, Y y) and the continuity of y∗( · ; z) give σ > 0

P-a.s. Moreover, from the continuity of y∗( · ; z) and y( · ; z) one gets that the set(x, y) ∈

Q | y(x; z) < y < y∗(x; z)

is open and not empty. These facts, combined with the fact

that y∗( · ; z) ≤ ϑ( · ; z) and with (4.22), imply that the last expression in (4.35) must be strictly

negative and we reach a contradiction. Therefore y( · ; z) ≥ y∗( · ; z) on D∗z ∩Dz. Since D

∗z ∩Dz =

(x ∨ x∗ , x ∧ x∗), this leads to x ≤ x∗ and x ≤ x∗ by monotonicity and continuity of y( · ; z) and

y∗( · ; z), hence D∗z ∩ Dz = (x∗ , x). Outside D∗

z ∩ Dz we then have y∗(x; z) = y ≤ y(x; z) for

x ≤ x∗ and y(x; z) = y ≥ y∗(x; z) for x ≥ x and the claim follows.

Case (ii): D∗z ∩ Dz = ∅. By monotonicity of y∗( · ; z) and y( · ; z) one has either x ≤ x∗ or

x ≥ x∗. If x ≤ x∗ then y( · ; z) ≥ y∗( · ; z) on I1; if x ≥ x∗ we can use the same arguments as

above to find x = x which contradicts the assumption that Dz 6= ∅.

Step 2. Here we show that y( · ; z) ≤ y∗( · ; z). Assume, by contradiction, that there exists

x ∈ I1 such that y(x; z) > y∗(x; z). Take y ∈ (y∗(x; z) , y(x; z)) and consider the stopping time

τ∗ = τ∗(x, y; z) := inft ≥ 0 | Y yt ≤ y∗(Xx

t ; z). This is the first optimal stopping time for the

problem (3.2), as it is the first entry time in the stopping region Az (cf. (3.46) and (4.2)). As in

Step 1 above, (3.41) and (4.27) give

E

[e−rτ∗v(Xx

τ∗ , Yyτ∗ ; z)

]= v(x, y; z) − E

[∫ τ∗

0e−rtcz(X

xt , z) dt

], (4.36)

E

[e−rτ∗w(Xx

τ∗ , Yyτ∗ ; z)

]= w(x, y; z) − E

[∫ τ∗

0e−rtH(Xx

t , Yyt ; z) dt

]. (4.37)

The Optimal Boundary of an Irreversible Investment Problem 26

By using (3.43) and a localization argument as in the proof of Theorem 3.19, we obtain

E[e−rτ∗v(Xx

τ∗ , Yyτ∗ ; z)

]= −E

[e−rτ∗Y y

τ∗

]. On the other hand, we know from Step 1 above that

y( · ; z) ≥ y∗( · ; z), hence E[e−rτ∗w(Xx

τ∗ , Yyτ∗ ; z)

]= −E

[e−rτ∗Y y

τ∗

]by (A-1), (A-3), the fact that

y is a natural boundary point and by localization arguments as in the proof of Lemma A.1.

Taking also into account that v ≥ w (cf. Lemma A.1) and subtracting (4.37) from (4.36) we

obtain

0 ≥ E

[∫ τ∗

0e−rt

(H(Xx

t , Yyt ; z) − cz(X

xt , z)

)dt

]

= −E

[∫ τ∗

0e−rt (cz(X

xt , z) + (rY y

t − µ2(Yyt )))1y∗(Xx

t ;z)<Yyt <y(Xx

t ;z)dt

]. (4.38)

Now τ∗ > 0 P-a.s. by continuity of trajectories of (Xx, Y y) and of y∗( · ; z). Moreover the set(x, y) ∈ Q | y∗(x; z) < y < y(x; z)

is open in Q and not empty, by continuity of y∗( · ; z)

and y( · ; z). Since by assumption y( · ; z) ≤ ϑ( · ; z), these facts together with (4.22) imply

that the last term in (4.38) must be strictly positive thus leading to a contradiction. Hence

y( · ; z) ≤ y∗( · ; z).

Remark 4.11. Expressions similar to (4.7) and (4.31) for the value function of optimal stopping

problems and their free-boundaries have already been proved in the context of numerous examples

with one dimensional diffusions and finite time-horizon (cf. [35] for a survey). However, to

the best of our knowledge, in the context of infinite time-horizon and genuine 2-dimensional

diffusions (4.7) and (4.31) are a novelty in the literature on their own.

Regarding the assumptions Cz 6= ∅ and Az 6= ∅ in Theorem 4.10, we provide the following

characterization.

Proposition 4.12. 1. The continuation set Cz is not empty if and only if the set

L+z :=

(x, y) ∈ Q | cz(x, z)− µ2(y) + ry > 0

(4.39)

is not empty.

2. The stopping set Az is not empty if and only if

limx↑x

E

[∫ ∞

0e−rtcz(X

xt , z)dt

]< −y. (4.40)

Proof. For the first claim notice that L+z ⊂ Cz (cf. also (3.34)) so that L+

z 6= ∅ ⇒ Cz 6= ∅. To

prove the reverse implication it suffices to observe that, by using (3.7) into (3.1), if L+z = ∅ then

any stopping rule would produce a payoff smaller or equal than the one of immediate stopping

and therefore Cz = ∅.

For the second claim we observe that

Az = ∅ ⇐⇒ Cz = Q ⇐⇒ τ∗ = +∞ P− a.s. ∀(x, y) ∈ Q ⇐⇒ v(x, y; z) > −y ∀(x, y) ∈ Q.

The Optimal Boundary of an Irreversible Investment Problem 27

Hence Az = ∅ if and only if

v(x, y; z) = E

[∫ ∞

0e−rtcz(X

xt , z)dt

]> −y ∀(x, y) ∈ Q. (4.41)

Then (4.40) implies that Az 6= ∅. Conversely, if Az 6= ∅, then there exists a point (x, y) ∈ Q such

that stopping at once is more profitable than (for instance) never stopping. For such a point

0 = y + v(x, y; z) ≥ y + E

[∫ ∞

0e−rtcz(X

xt , z)dt

]. (4.42)

Since y > y and cz( · , z) is nonincreasing (cf. Assumption 2.2-(ii)), then (4.40) must hold.

In principle Theorem 4.10 fully characterizes the optimal boundary of problem (3.2), but it

has the drawback that the region D∗z = (x∗, x

∗), with x∗ and x∗ as in (4.30), is defined implicitly.

For the purpose of numerical evaluation of (4.31) it would be helpful to know D∗z in advance

rather than computing it at the same time as y∗( · ; z). Recall (4.23) and define

θ∗ := xϑ(·;z) = infx ∈ I1 |ϑ(x; z) > y

, θ∗ := xϑ(·;z) = sup

x ∈ I1 | ϑ(x; z) < y

, (4.43)

with the convention inf ∅ = x, sup ∅ = x. Since y∗( · ; z) ≤ ϑ( · ; z), we have x∗ ≥ θ∗ and x∗ ≥ θ∗.

To characterize x∗ we will make use of the following algebraic equation

−y =

∫ ∞

0e−rt

(∫ x

x

p1(t, x; ξ)cz(ξ, z)dξ − ry

∫ x

x

p1(t, x, ξ)dξ)dt. (4.44)

Similarly, if y < +∞, a characterization of x∗ will be given in terms of the algebraic equation

−y =

∫ ∞

0e−rt

(∫ x

x

p1(t, x; ξ)cz(ξ, z)dξ − ry

∫ x

x

p1(t, x, ξ)dξ)dt. (4.45)

Proposition 4.13. Let Assumptions 4.2, 4.5, 4.7 hold. Let Cz 6= ∅ and Az 6= ∅. Then

1. x∗ ∈ I1 if and only if (4.44) has a unique solution x ∈ (θ∗, x). Moreover x∗ = x and if

such solution does not exist then x∗ = x.

2. If y < +∞, then x∗ ∈ I1 if and only if (4.45) has a unique solution x′ ∈ (θ∗, x). Moreover

x∗ = x′ and if such solution does not exist, then x∗ = x.

3. If y = +∞ and there exists λ > 0 such that r − ∂µ2

∂y≥ λ on I2, then x

∗ = x.

Proof. 1. Existence and uniqueness of a solution of (4.44) (θ∗, x) is discussed in Appendix

A.2.

Proof of ⇒. Take a sequence xn, n ∈ N ⊂ I1 such that xn ↓ x∗ and notice that by

Theorem 4.10 we have for every n ∈ N

−y∗(xn; z) =

∫ ∞

0e−rt

[ ∫ x

x

p1(t, xn, ξ)cz(ξ, z)

(∫ y

y∗(ξ;z)p2(t, y

∗(xn; z), η)dη

)dξ

]dt (4.46)

−

∫ ∞

0e−rt

[ ∫ x

x

p1(t, xn, ξ)

(∫ y∗(ξ;z)

y

(rη − µ2(η))p2(t, y∗(xn; z), η)dη

)dξ

]dt.

The Optimal Boundary of an Irreversible Investment Problem 28

We aim to take limits of (4.46) as n ↑ ∞. For the left hand-side of (4.46) we have y∗(xn; z) ↓ y,

by continuity of y∗( · ; z) and definition of x∗. On the other hand, taking into account that

y∗( · ; z) = y for ξ ≤ x∗, the first term of the right-hand side of (4.46) can be written as

∫ ∞

0e−rt

[ ∫ x

x

p1(t, xn, ξ)cz(ξ, z)

(∫ y

y∗(ξ;z)p2(t, y

∗(xn; z), η)dη

)dξ

]dt (4.47)

=

∫ ∞

0e−rt

[ ∫ x∗

x

p1(t, xn, ξ)cz(ξ, z)dξ

+

∫ x

x∗

p1(t, xn, ξ)cz(ξ, z)

(∫ y

y

1η>y∗(ξ;z)p2(t, y∗(xn; z), η)dη

)dξ

]dt.

Now notice that:

(i) for any t > 0 the sequence of probability measures with densities p1(t, xn, ξ), n ∈ N on

I1 converges pointwisely to p1(t, x∗, ξ)dξ by Assumption 3.13;

(ii) for any given and fixed t > 0 and z ∈ R+ the sequence of probability measures with

densities p2(t, y∗(xn; z), η), n ∈ N on I2 converges weakly to the Dirac’s delta measure δy(η),

due to #8 of [31, Ch. II, Sec. 3] (see also (A-9) in Appendix A);

(iii) for every ξ > x∗, the function I2 → R, η 7→ cz(ξ, z)1η>y∗(ξ;z) ≡ 0 δy-a.e.

Then, taking into account (i)-(iii) we can apply Portmanteau Theorem to the integral with

respect to dη in the right hand side of (4.47) and dominated convergence to the one with respect

to dξ to obtain

limn→+∞

∫ x

x

p1(t, xn, ξ)cz(ξ, z)

(∫ y

y∗(ξ;z)p2(t, y

∗(xn; z), η)dη

)dξ =

∫ x∗

x

p1(t, x∗, ξ)cz(ξ, z)dξ

Finally, a further application of dominated convergence to the integral with respect to dt, gives

limn→+∞

∫ ∞

0e−rt

[∫ x

x

p1(t, xn, ξ)cz(ξ, z)

(∫ y

y∗(ξ;z)p2(t, y

∗(xn; z), η)dη

)dξ

]dt

=

∫ ∞

0e−rt

[∫ x∗

x

p1(t, x∗, ξ)cz(ξ, z)dξ

]dt.

Similar arguments can be applied to the second term of the right-hand side of (4.46). In fact

for ξ > x∗ the map η 7→ (rη − µ2(η))1η≤y∗(ξ;z) is bounded on I2 and it is continuous at y.

Moreover (rη − µ2(η))1η≤y∗(ξ;z) = ry − µ2(y), δy-a.e.

Proof of ⇐. Assume now that θ∗ < x and that x ∈ (θ∗, x) uniquely solves (4.44). It is

proven in Appendix A, Section A.2, that x is the optimal boundary of the one-dimensional

optimal stopping problem

v(x; z) := supτ∈T

E

[∫ τ

0e−rtcz(X

xt , z)dt− ye−rτ

], (4.48)

and hence that Az := x ∈ I1 | v(x; z) = −y = x ∈ I1 |x ≥ x. By arguments as in the proof

of Proposition 3.7 we have v(x; z) = limy↓y v(x, y; z). Moreover 0 < v(x; z) + y ≤ v(x, y; z) + y

The Optimal Boundary of an Irreversible Investment Problem 29

for all (x, y) ∈ (x, x)× I2 by monotonicity of y 7→ v(x, y; z) + y (cf. Proposition 4.3), and hence

x∗ ≥ x > x. Also x∗ < x, since otherwise Az = ∅ thus contradicting the assumption Az 6= ∅.

Therefore x∗ ∈ I1 and hence by the arguments of the first part of this proof x∗ solves (4.44).

Since such solution is unique it must be x = x∗.

2. The proof of this second claim works thanks to arguments similar to the ones employed

for the first one. One has to consider, in place of (4.48), the optimal stopping problem

v(x; z) := supτ∈T

E

[∫ τ

0e−rtcz(X

xt , z)dt− ye−rτ

].

3. The further assumption guarantees that ϑ( · ; z) < +∞ on I1 and the claim follows.

Remark 4.14. Despite their rather involved definition x∗ and x∗ have a quite clear probabilistic

interpretation. In fact, they are the free-boundaries of the optimal stopping problems

v(x; z) := supτ∈T

E

[∫ τ

0e−rtcz(X

xt , z)dt − ye−rτ

], v(x; z) := sup

τ∈TE

[∫ τ

0e−rtcz(X

xt , z)dt− ye−rτ

],

respectively, with v( · ; z) = limy↓y v( · , y; z) and v( · ; z) = limy↑y v( · , y; z).

5 The Optimal Control

In this section we characterize the optimal control ν∗ of (2.13) by showing that it is optimal

to exert the minimal effort needed to reflect the (optimally controlled) state process Zz,ν∗ at a

(random) boundary intimately connected to y∗ of Theorem 4.10.

5.1 The action/inaction regions

Define

C := (x, y, z) ∈ O | v(x, y; z) > −y and A := (x, y, z) ∈ O | v(x, y; z) = −y. (5.1)

The sets C and A are respectively the candidate inaction region and the candidate action region

for the control problem (2.13).

Remark 5.1. We notice that the formal connection (3.4) yields

C = (x, y, z) ∈ O | Vz(x, y, z) > −y, A = (x, y, z) ∈ O | Vz(x, y, z) = −y. (5.2)

Intuitively, A is the region in which it is optimal to invest immediately, whereas C is the region

in which it is profitable to delay the investment option.

Throughout this section all the assumptions made so far will be standing assumptions, i.e.

Assumptions 2.1, 2.2, 2.3, 3.3, 3.13, 4.2, 4.5 and 4.7 hold and we will not repeat them in the

statement of the next results.

It immediately follows from the fact that cz(x, ·) is nondecreasing for each x ∈ I1 that

The Optimal Boundary of an Irreversible Investment Problem 30

Proposition 5.2. The function z 7→ v(x, y; z) is nondecreasing for every (x, y) ∈ Q.

The nondecreasing property of z 7→ v(x, y; z) implies that for fixed (x, y) ∈ Q the region A

is below C, and we define the boundary between these two regions by

z∗(x, y) := infz ∈ R+ | v(x, y; z) > −y, (5.3)

with the convention inf ∅ = ∞. Then (5.1) can be equivalently written as

C = (x, y, z) ∈ O | z > z∗(x, y), A = (x, y, z) ∈ O | z ≤ z∗(x, y). (5.4)

We can also easily observe from (4.1) and (5.3) and from the nondecreasing property of z 7→

v(x, y; z) and of y 7→ v(x, y; z) + y (cf. Proposition 5.2 and Proposition 4.3, respectively) that

z > z∗(x, y) ⇐⇒ v(x, y; z) > −y ⇐⇒ y > y∗(x; z), (x, y, z) ∈ O. (5.5)

Hence for any x ∈ I1, z∗ of (5.3) can be seen as the pseudo-inverse of the nonincreasing (cf.

Proposition 4.4) function z 7→ y∗(x; z); that is,

z∗(x, y) = infz ∈ R+ | y > y∗(x; z), (x, y) ∈ Q. (5.6)

It thus follows that the characterization of y∗ of Theorem 4.10 is actually equivalent to a complete

characterization of z∗ thanks to (5.6).

Set

z(x, y) := infz ∈ R+ | cz(x, z)− µ2(y) + ry > 0, (x, y) ∈ Q,

with the usual convention inf ∅ = ∞, and recall ϑ(x; z) of Lemma 4.9. Then the nondecreasing

property of z 7→ cz(x, z) − µ2(y) + ry and of y 7→ cz(x, z) − µ2(y) + ry (cf. Assumption 2.2 and

Assumption 4.2, respectively) implies that

z > z(x, y) ⇐⇒ cz(x, z) − µ2(y) + ry > 0 ⇐⇒ y > ϑ(x; z), (x, y, z) ∈ O,

and therefore that

z(x, y) = infz ∈ R+ | y > ϑ(x; z). (5.7)

Proposition 5.3. One has

1. z∗ ≤ z over Q.

2. z∗( · , y) is nondecreasing for each y ∈ I2 and z∗(x, · ) is nonincreasing for each x ∈ I1.

3. z∗( · , y) is right-continuous for each y ∈ I2 and z∗(x, · ) is left-continuous for each x ∈ I1.

4. (x, y) 7→ z∗(x, y) is upper-semicontinuous.

The Optimal Boundary of an Irreversible Investment Problem 31

Proof. 1. It follows by (5.6), (5.7) and (3.34).

2. The first claim follows from the fact that v( · , y; z) is nonincreasing for each y ∈ I2,

z ∈ R+, by Proposition 3.6; the fact that y 7→ v(x, y; z) + y is nondecreasing for each x ∈ I1,

z ∈ R+ (cf. proof of Proposition 4.3) implies the second one.

3. The proof of these two properties follows from the fact that v(·) is continuous by Proposi-

tion 3.7 and Remark 3.8, and from point 2 above by using arguments as those employed in [25,

Prop. 2.2].

4. Notice that by (5.5) one has

(x, y) ∈ I1 × I2 : z > z∗(x, y) = (x, y) ∈ I1 × I2 : v(x, y; z) > −y, (5.8)

for any z ∈ R+. The set on the right-hand side above is open since it is the preimage of an open

set via the continuous mapping (x, y) 7→ v(x, y; z) + y (cf. Proposition 3.7). Hence the set on

the left-hand side of (5.8) is open as well and thus (x, y) 7→ z∗(x, y) is upper-semicontinuous.

Now Proposition 5.3 and the following

Assumption 5.4. limz↑∞

cz(x, z) = ∞ for every x ∈ I1

imply

Proposition 5.5. Under Assumption 5.4, z is finite on Q.

Then, thanks to Proposition 5.3-(1) one also has

Corollary 5.6. z∗ is finite on Q.

The topological characterization of the regions C and A is given in the following

Proposition 5.7. C is open and A is closed. Moreover, under Assumption 5.4, they are con-

nected.

Proof. The fact that C is open and A is closed follows from (5.1) and Remark 3.8. Corollary

5.6 and (5.4) imply the second part of the claim.

5.2 Optimal Control: a Verification Theorem

The results obtained in Section 3 on the optimal stopping problem (3.2) (especially the super-

harmonic characterization of Theorem 3.19) allow us to provide the expression of the optimal

control ν∗ of problem (2.13) in terms of the boundary z∗ of (5.3). Moreover, as a byproduct,

we will also show (see Corollary 5.10 below) that the connection (3.4) holds true with V as in

(2.13) and v as in (3.2).

Recall (3.2) and define the functions

Φ(x, z) := E

[ ∫ ∞

0e−rtc(Xx

t , z)dt

], (x, z) ∈ I1 × R

+, (5.9)

The Optimal Boundary of an Irreversible Investment Problem 32

ϕ(x, z) :=∂

∂zΦ(x, z) = E

[ ∫ ∞

0e−rtcz(X

xt , z)dt

], (x, z) ∈ I1 × R

+, (5.10)

and

U(x, y, z) := Φ(x, z)−

∫ ∞

z

(v(x, y; q) − ϕ(x, q))dq, (x, y, z) ∈ O. (5.11)

Notice that v(x, y; z) ≥ ϕ(x, z) for every (x, y, z) ∈ O, and therefore function U in (5.11) above

is well-defined (but, a priori, it may be equal to −∞).

Introduce the nondecreasing process

ν∗t := sup0≤s≤t

[z∗(Xxs , Y

ys )− z]+, t ≥ 0, ν∗0− = 0, (5.12)

with z∗(x, y) as in (5.3).

Proposition 5.8. Under Assumption 5.4 the process ν∗ of (5.12) is an admissible control.

Proof. Recall the set of admissible controls V of (2.3). Clearly ν∗ is a.s. finite thanks to

Corollary 5.6. To prove that ν∗ ∈ V it remains to show that: i) t 7→ ν∗t is right-continuous with

left-limits; ii) ν∗ is (Ft)-adapted.

We start by proving i). Clearly, t 7→ ν∗t admits left-limit at any point since it is nondecreasing.

To show that ν∗ has right-continuous sample paths, first notice that

lim sups↓t

z∗(Xxs , Y

ys ) ≤ z∗(Xx

t , Yyt ) (5.13)

by upper-semicontinuity of z∗ (cf. Proposition 5.3) and continuity of (Xx· , Y

y· ). Moreover, from

(5.12) and (5.13) we obtain

lims↓t

ν∗s = ν∗t ∨ lims↓t

supt<u≤s

[z∗(Xxu , Y

yu )− z]+

= ν∗t ∨ lim sups↓t

[z∗(Xxs , Y

ys )− z]+ ≤ ν∗t ∨ [z∗(Xx

t , Yyt )− z]+ = ν∗t . (5.14)

Since lims↓t ν∗s ≥ ν∗t by monotonicity of t 7→ ν∗t , then (5.14) implies right continuity.

As for ii) the process z∗(Xx, Y y) is progressively measurable since it is the composition of

the the Borel-measurable function z∗ (which is upper semicontinuous by Proposition 5.3) with

the progressively measurable process (Xx, Y y). Therefore ν∗ is progressively measurable by [14,

Th. IV.33, part (a)], hence adapted and ii) above holds.

Theorem 5.9. Let Assumption 5.4 hold. Fix (x, y, z) ∈ O and take Φ(x, z), ϕ(x, z) and U(x, z)

as in (5.9), (5.10) and (5.11), respectively. Then one has U(x, y, z) = V (x, y, z) and ν∗ as in

(5.12) is optimal for the singular control problem (2.13).

It clearly follows from Theorem 5.9 the following

Corollary 5.10. The identity (3.4) holds true.

The Optimal Boundary of an Irreversible Investment Problem 33

The proof of Theorem 5.9 is inspired by the arguments developed in [1] and [16]. It is

based on a probabilistic verification argument relying on the superharmonic characterization of

v described in Theorem 3.19.

Proof of Theorem 5.9. For ν ∈ V define its right-continuous inverse (cf. [38, Ch. 0, Sec. 4])

τν(ξ) := inft ≥ 0 | νt > ξ, ξ ≥ 0. (5.15)

The process τν := τν(ξ), ξ ≥ 0 has increasing, right-continuous sample paths and hence it

admits left-limits

τν−(ξ) := inft ≥ 0 | νt ≥ ξ, ξ ≥ 0. (5.16)

The set of points ξ ∈ R+ at which τν(ξ)(ω) 6= τν−(ξ)(ω) is a.s. countable for a.e. ω ∈ Ω.

Since ν is right-continuous and τν(ξ) is the first entry time of an open set, it is an (Ft+)-

stopping time for any given and fixed ξ ≥ 0. However (Ft)t≥0 is right-continuous (cf. Section

2) and hence τν(ξ) is an (Ft)-stopping time. Moreover, τν−(ξ) is the first entry time of the

right-continuous process ν into a closed set and hence it is an (Ft)-stopping time as well for any

ξ ≥ 0. It then follows by (3.45) that

v(x, y; q) ≥ E

[e−rτν(ξ)v(Xx

τν (ξ), Yyτν(ξ); q) +

∫ τν(ξ)

0e−rscz(X

xs , q)ds

], (5.17)

for any ξ ≥ 0 and (x, y, q) ∈ O. Then for any (x, y, z) ∈ O, taking ξ = q− z, q ≥ z in (5.17) and

recalling (5.9), (5.10) and (5.11) we obtain

U(x, y, z)− Φ(x, z) ≤ −

∫ ∞

z

(E

[e−rτν(q−z)v(Xx

τν (q−z), Yyτν(q−z); q) +

+

∫ τν(q−z)

0e−rscz(X

xs , q)ds

])dq +

∫ ∞

z

E

[ ∫ ∞

0e−rscz(X

xs , q)ds

]dq

≤

∫ ∞

z

E

[e−rτν(q−z)Y y

τν(q−z)

]dq −

∫ ∞

z

E

[ ∫ τν(q−z)

0e−rscz(X

xs , q)ds

]dq

+

∫ ∞

z

E

[ ∫ ∞

0e−rscz(X

xs , q)ds

]dq, (5.18)

where we have used that v( · , ζ; · ) ≥ −ζ (cf. Proposition 3.6) in the second inequality. We

now claim (and we will prove it later) that we can apply Fubini-Tonelli’s Theorem in the last

expression of (5.18) to obtain

U(x, y, z) − Φ(x, z) ≤ E

[ ∫ ∞

z

e−rτν(q−z)Y yτν(q−z)dq −

∫ ∞

z

(∫ τν(q−z)

0e−rscz(X

xs , q)ds

)dq

]

+E

[ ∫ ∞

z

(∫ ∞

0e−rscz(X

xs , q)ds

)dq

]. (5.19)

The change of variable formula of [38, Ch. 0, Prop. 4.9] (see also [1, eq. (4.7)]) implies∫ ∞

z

e−rτν(q−z)Y yτν(q−z)dq =

∫ ∞

0e−rsY y

s dνs. (5.20)

The Optimal Boundary of an Irreversible Investment Problem 34

Moreover τν(q − z) < s if and only if νs > q − z, s ≥ 0 and therefore from (5.19) and (5.20) we

obtain

U(x, y, z) − Φ(x, z) ≤ E

[ ∫ ∞

0e−rsY y

s dνs +

∫ ∞

z

(∫ ∞

τν(q−z)e−rscz(X

xs , q)ds

)dq

]

= E

[ ∫ ∞

0e−rsY y

s dνs +

∫ ∞

z

(∫ ∞

0e−rscz(X

xs , q)1νs>q−zds

)dq

]

= E

[ ∫ ∞

0e−rsY y

s dνs +

∫ ∞

0e−rs

(∫ z+νs

z

cz(Xxs , q)dq

)ds

](5.21)

= E

[ ∫ ∞

0e−rsY y

s dνs +

∫ ∞

0e−rs

[c(Xx

s , Zz,νs )− c(Xx

s , z)]ds

]

= Jx,y,z(ν)− Φ(x, z).

Since ν ∈ V is arbitrary it follows

U(x, y, z) ≤ V (x, y, z). (5.22)

Now we want to show that picking ν∗ as in (5.12) in the arguments above all the inequalities

become equalities due to (3.46). First notice that (3.46), (5.1) and (5.4) give

τ∗(x, y; q) = inft ≥ 0 | z∗(Xxt , Y

yt ) ≥ q. (5.23)

Then fix z ∈ R+, take t ≥ 0 arbitrary and note that by (5.16) and (5.23) we have P-a.s. the

equivalences

τν∗

− (q − z) ≤ t ⇐⇒ ν∗t ≥ q − z ⇐⇒ sup0≤s≤t

[z∗(Xxs , Y

ys )− z]+ ≥ q − z

⇐⇒ z∗(Xxθ , Y

yθ ) ≥ q for some θ ∈ [0, t] ⇐⇒ τ∗(x, y; q) ≤ t.

So we can conclude that τν∗

− (q − z) = τ∗(x, y; q) P-a.s. and for a.e. q ≥ z. However, by (5.15)

and (5.16) we also have τν∗

− (q − z) = τν∗(q − z) P-a.s. and for a.e. q ≥ z; hence

τν∗

(q − z) = τ∗(x, y; q) P-a.s. and for a.e. q ≥ z. (5.24)

Now take ν = ν∗ and ξ = q − z in order to obtain equality in (5.17) by Theorem 3.19 and

(5.24). Optimality of τ∗ = τν∗(cf. (5.24)) also gives equality in (5.18); then we can interchange

the integrals and argue as in (5.19) and (5.21) to obtain U(x, y, z) = Jx,y,z(ν∗). Then U = V

on O by (5.22) and ν∗ is optimal.

To conclude the proof we need to show that we could actually interchange the order of

integration in (5.18) to get (5.19). Clearly∫ ∞

z

E

[e−rτν(q−z)Y y

τν(q−z)

]dq = E

[ ∫ ∞

z

e−rτν(q−z)Y yτν(q−z)dq

],

by Tonelli’s Theorem since Y y has positive sample paths. Therefore we have only to show that

E

[ ∫ ∞

z

(∫ ∞

τν(q−z)e−rs|cz(X

xs , q)|ds

)dq

]<∞. (5.25)

The Optimal Boundary of an Irreversible Investment Problem 35

Define

q∗s := infq ∈ R : cz(Xxs , q) > 0,

which exists unique since c(x, ·) is convex. Now, recall that τν(q−z) < s if and only if νs > q−z,

s ≥ 0; then Tonelli’s Theorem, Remark 2.7 and the fact that c ≥ 0 give

E

[ ∫ ∞

z

( ∫ ∞

τν(q−z)e−rs|cz(X

xs , q)|ds

)dq

]= E

[∫ ∞

z

( ∫ ∞

0e−rs|cz(X

xs , q)|1τν (q−z)<sds

)dq

]

= E

[ ∫ ∞

0e−rs

( ∫ z+νs

z

|cz(Xxs , q)|dq

)ds

]= E

[∫ ∞

0e−rs

( ∫ z+νs

(z+νs)∧q∗s

cz(Xxs , q)dq

)ds

]

−E

[ ∫ ∞

0e−rs

( ∫ (z+νs)∧q∗s

z

cz(Xxs , q)dq

)ds

]

≤ E

[ ∫ ∞

0e−rsc(Xx

s , z)ds +

∫ ∞

0e−rsc(Xx

s , z + νs)ds

]<∞.

A Appendix

Lemma A.1. Let Assumptions 4.2, 4.5, 4.7 hold and assume that Cz 6= ∅ and Az 6= ∅. Let

y( · ; z) : I1 → I2 be a nontrivial solution of (4.31) and take w as in (4.25). Then v( · ; z) ≥

w( · ; z) on Q.

Proof. Recall the notation introduced in (4.29).

Step 1. Since y( · ; z) is a nontrivial solution of (4.31), i.e. of (4.32), it is easy to see that w

of (4.25) verifies

w(x, y(x; z); z) = −y(x; z), ∀x ∈ Dz, (A-1)

and therefore

w(x, y(x; z); z) ≤ v(x, y(x; z); z), ∀x ∈ Dz, (A-2)

Step 2. Here we show that

w(x, y; z) = −y, ∀y < y(x; z), ∀x ∈ Dz ∪ [x, x). (A-3)

which implies

w(x, y; z) ≤ v(x, y; z), ∀y < y(x; z), x ∈ Dz ∪ [x, x).

Take x ∈ Dz ∪ [x, x), y < y(x; z) and define σ = σ(x, y; z) := inft ≥ 0 |Y y

t ≥ y(Xxt ; z)

. By

definition of y( · ; z) and σ we have

H(Xxt , Y

yt ; z) = −

(rY y

t − µ2(Yyt )

), ∀t ≤ σ, P-a.s. (A-4)

The Optimal Boundary of an Irreversible Investment Problem 36

Then using the martingale property (4.27) up to the stopping time σ ∧ n, n ∈ N, it follows by

(A-4) that

w(x, y; z) = E

[− e−rσY y

σ 1σ≤n + e−rnw(Xxn , Y

yn ; z)1σ>n −

∫ σ∧n

0e−rt

(rY y

t − µ2(Yyt )

)dt

].

(A-5)

Assumption 2.3, (4.24) and the bound (4.26), give in the limit as n→ ∞

w(x, y; z) = E

[− e−rσY y

σ −

∫ σ

0e−rt

(rY y

t − µ2(Yyt )

)dt

]= y, (A-6)

where the last equality follows by Lemma 3.5. Hence (A-3) is proved.

Step 3. Here we prove that

w(x, y; z) ≤ v(x, y; z), ∀y > y(x; z), ∀x ∈ (x, x] ∪ Dz. (A-7)

Take x ∈ (x, x] ∪ Dz and y > y(x; z) and consider the stopping time

τ = τ(x, y; z) := inft ≥ 0 | Y y

t ≤ y(Xxt ; z)

.

By definitions of y( · ; z) and τ and by using the same localization argument as in Step 2 above,

we obtain

w(x, y; z) = E

[−e−rτY y

τ +

∫ τ

0e−rscz(X

xt , z)dt

]≤ v(x, y; z). (A-8)

Step 4. Now Lemma A.1 follows by (A-1), (A-3) and (A-7).

A.1 Further properties of natural boundaries

Here we show that µ2(y) = σ2(y) = 0. The same holds for y if it is finite. Analogously, µ1 and

σ1 are zero at x and x whenever the latter are finite. For the proof we rely on #8 of [31, Ch. II,

Sec. 3, p. 32] that guarantees

limy↓y

E

[∫ ∞

0e−rtf(Y y

t )dt

]=

1

rf(y) for any f ∈ Cb(R); (A-9)

that is, the family of probability measures on I2 with densities p2(t, y, ·), y ∈ I2, t > 0, (cf.

Assumption 3.13) converges weakly to the Dirac’s delta measure δy(·), for any t > 0, when y ↓ y.

Case 1. If I2 is bounded, an application of Dynkin’s formula to any g ∈ C2b (R) leads to

g(y) = −E

[ ∫ ∞

0e−rt

(12σ22(Y

yt )g

′′(Y yt ) + µ2(Y

yt )g

′(Y yt )− rg(Y y

t ))dt

]. (A-10)

The Optimal Boundary of an Irreversible Investment Problem 37

Then taking limits as y ↓ y, noting that µ2 and σ2 are bounded and continuous and by applying

(A-9) we get

1

2σ22(y)g

′′(y) + µ2(y)g′(y) = 0, (A-11)

and since g is arbitrary it must be µ2(y) = σ2(y) = 0.

Case 2. If I2 is unbounded (i.e. if I2 = (y,∞)) we approximate (µ2, σ2) by continuous

bounded functions (µn2 , σn2 ) such that µn2 = µ2 and σn2 = σ2 on [y, n ∨ y] with µn2 (y) → µ2(y)

and σn2 (y) → σ2(y) as n → ∞ pointwise on I2. For y ∈ (y, n ∨ y) the associated diffusion

with coefficients µn2 and σn2 , denoted by Y y,n, coincides with Y y up to the first exit time from

(y, n ∨ y) by uniqueness of the solution of (2.2); moreover, y is a natural boundary for Y y,n as

well. Repeating arguments as in Case 1 above we get µn2 (y) = σn2 (y) = 0 for all n ∈ N, thus

µ2(y) = σ2(y) = 0.

A.2 Discussion on Problem (4.48)

Problem (4.48) is standard in the optimal stopping literature (cf. for instance [35] for methods

of solution) and hence we only sketch arguments leading to its main properties. It is easy to see

that x 7→ v(x; z) is nonincreasing and hence there exists b∗ ∈ I1 such that Az = [b∗, x), where

the boundary value x cannot be included as otherwise Az = ∅ thus contradicting the assumption

of Proposition 4.13. It is possible to show that v( · ; z) ∈ C1(I1), vxx( · ; z) is locally bounded at

b∗ and hence that the probabilistic representation

v(x; z) = E

[ ∫ ∞

0e−rt

(cz(X

xt ; z)1Xx

t <b∗ − ry1Xxt ≥b∗

)dt]

(A-12)

holds by Ito-Tanaka formula. Since (A-12) holds for any x ∈ I1, then if b∗ ∈ I1 by evaluating

(A-12) for x = b∗, one easily finds that b∗ solves (4.44). Arguments similar to (but simpler than)

those employed in the proof of Theorem 4.10 show that (4.44) admits a unique solution in (θ∗, x)

and therefore it must be x = b∗. On the other hand, if b∗ = x, repeating arguments as those of

the proof of Theorem 4.10, Step 2, one can show that x = b∗, thus concluding.

Acknowledgments. The authors thank Goran Peskir, Frank Riedel and Mauro Rosestolato

for useful suggestions and references.

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